id string | topic string | difficulty int64 | problem_statement string | solution_paths list | reconciliation dict | error_catalogue list | conceptual_takeaway string |
|---|---|---|---|---|---|---|---|
math-001901 | Logic: Quantifiers — Negation Rules | 1 | Solve with verification: Negate the statement and simplify (push negation inward) using quantifier laws:
$$\forall x\in\mathbb{R}\ \exists y\in\mathbb{R}\text{ such that } y>x+47.$$
(a) Give the symbolic negation step-by-step.
(b) Translate both the original statement and its negation into plain English to confirm the... | [
{
"method_name": "English Semantics Cross-Check",
"approach": "Translate to English, negate carefully, then translate back to symbols.",
"steps": [
"Step 1: Original: 'For every real $x$, there exists a real $y$ that is greater than $x+47$.'",
"Step 2: Negation: 'There exists a real $x$ such... | {
"consistency_check": "Both approaches agree after simplification. Final answer: $\\boxed{\\exists x\\in\\mathbb{R}$.\nThe formal quantifier-negation rules and the English semantics yield the same statement: the existence of an $x$ that upper-bounds all reals after shifting by a constant.",
"robustness_analysis": ... | [
{
"error_description": "Negated only the inequality but kept quantifiers unchanged.",
"why_plausible": "The inequality is the most visible part of the statement, so attention focuses there.",
"why_wrong": "Quantifiers determine scope; the correct negation flips $\\forall\\leftrightarrow\\exists$ and cha... | Remember: Negating quantified statements requires flipping each quantifier and negating the predicate while keeping variable order. Always sanity-check by translating to plain English. (Here the result is $\boxed{\exists x\in\mathbb{R}$.) |
math-001902 | Logic: Scope — Order of Quantifiers | 1 | Give an answer and a quick verification: Negate the statement and simplify (push negation inward) using quantifier laws:
$$\forall x\in\mathbb{R}\ \exists y\in\mathbb{R}\text{ such that } y>x+284.$$
(a) Give the symbolic negation step-by-step.
(b) Translate both the original statement and its negation into plain Engli... | [
{
"method_name": "English Semantics Cross-Check",
"approach": "Translate to English, negate carefully, then translate back to symbols.",
"steps": [
"Step 1: Original: 'For every real $x$, there exists a real $y$ that is greater than $x+284$.'",
"Step 2: Negation: 'There exists a real $x$ suc... | {
"consistency_check": "Both approaches agree after simplification. Final answer: $\\boxed{\\exists x\\in\\mathbb{R}$.\nThe formal quantifier-negation rules and the English semantics yield the same statement: the existence of an $x$ that upper-bounds all reals after shifting by a constant.",
"robustness_analysis": ... | [
{
"error_description": "Negated only the inequality but kept quantifiers unchanged.",
"why_plausible": "The inequality is the most visible part of the statement, so attention focuses there.",
"why_wrong": "Quantifiers determine scope; the correct negation flips $\\forall\\leftrightarrow\\exists$ and cha... | Key idea: Negating quantified statements requires flipping each quantifier and negating the predicate while keeping variable order. Always sanity-check by translating to plain English. |
math-001903 | Logic: Scope — Order of Quantifiers | 1 | Compute the requested quantity: Negate the statement and simplify (push negation inward) using quantifier laws:
$$\forall x\in\mathbb{R}\ \exists y\in\mathbb{R}\text{ such that } y>x+10.$$
(a) Give the symbolic negation step-by-step.
(b) Translate both the original statement and its negation into plain English to conf... | [
{
"method_name": "English Semantics Cross-Check",
"approach": "Translate to English, negate carefully, then translate back to symbols.",
"steps": [
"Step 1: Original: 'For every real $x$, there exists a real $y$ that is greater than $x+10$.'",
"Step 2: Negation: 'There exists a real $x$ such... | {
"consistency_check": "Cross-check: both derivations land on the same invariant quantity. Final answer: $\\boxed{\\exists x\\in\\mathbb{R}$.\nThe formal quantifier-negation rules and the English semantics yield the same statement: the existence of an $x$ that upper-bounds all reals after shifting by a constant.",
... | [
{
"error_description": "Negated only the inequality but kept quantifiers unchanged.",
"why_plausible": "The inequality is the most visible part of the statement, so attention focuses there.",
"why_wrong": "Quantifiers determine scope; the correct negation flips $\\forall\\leftrightarrow\\exists$ and cha... | Takeaway: Negating quantified statements requires flipping each quantifier and negating the predicate while keeping variable order. Always sanity-check by translating to plain English. |
math-001904 | Logic: Quantifiers — Negation Rules | 1 | Give a theorem-based solution: Negate the statement and simplify (push negation inward) using quantifier laws:
$$\forall x\in\mathbb{R}\ \exists y\in\mathbb{R}\text{ such that } y>x+23.$$
(a) Give the symbolic negation step-by-step.
(b) Translate both the original statement and its negation into plain English to confi... | [
{
"method_name": "English Semantics Cross-Check",
"approach": "Translate to English, negate carefully, then translate back to symbols.",
"steps": [
"Step 1: Original: 'For every real $x$, there exists a real $y$ that is greater than $x+23$.'",
"Step 2: Negation: 'There exists a real $x$ such... | {
"consistency_check": "Cross-check: both derivations land on the same invariant quantity. Final answer: $\\boxed{\\exists x\\in\\mathbb{R}$.\nThe formal quantifier-negation rules and the English semantics yield the same statement: the existence of an $x$ that upper-bounds all reals after shifting by a constant.",
... | [
{
"error_description": "Negated only the inequality but kept quantifiers unchanged.",
"why_plausible": "The inequality is the most visible part of the statement, so attention focuses there.",
"why_wrong": "Quantifiers determine scope; the correct negation flips $\\forall\\leftrightarrow\\exists$ and cha... | Core principle: Negating quantified statements requires flipping each quantifier and negating the predicate while keeping variable order. Always sanity-check by translating to plain English. (Here the result is $\boxed{\exists x\in\mathbb{R}$.) |
math-001905 | Logic: Quantifiers — Negation Rules | 1 | Indicate where a theorem is used: Negate the statement and simplify (push negation inward) using quantifier laws:
$$\forall x\in\mathbb{R}\ \exists y\in\mathbb{R}\text{ such that } y>x+89.$$
(a) Give the symbolic negation step-by-step.
(b) Translate both the original statement and its negation into plain English to co... | [
{
"method_name": "Quantifier Laws",
"approach": "Use $\\neg\\forall=\\exists\\neg$ and $\\neg\\exists=\\forall\\neg$, then negate the inequality.",
"steps": [
"Step 1: Start: $\\neg(\\forall x\\ \\exists y\\ P(x,y))$.",
"Step 2: Apply $\\neg\\forall x\\ Q(x)\\equiv \\exists x\\ \\neg Q(x)$ t... | {
"consistency_check": "Cross-check: both derivations land on the same invariant quantity. Final answer: $\\boxed{\\exists x\\in\\mathbb{R}$.\nThe formal quantifier-negation rules and the English semantics yield the same statement: the existence of an $x$ that upper-bounds all reals after shifting by a constant.",
... | [
{
"error_description": "Negated only the inequality but kept quantifiers unchanged.",
"why_plausible": "The inequality is the most visible part of the statement, so attention focuses there.",
"why_wrong": "Quantifiers determine scope; the correct negation flips $\\forall\\leftrightarrow\\exists$ and cha... | Core principle: Negating quantified statements requires flipping each quantifier and negating the predicate while keeping variable order. Always sanity-check by translating to plain English. (Here the result is $\boxed{\exists x\in\mathbb{R}$.) |
math-001906 | Logic: Scope — Order of Quantifiers | 1 | State any required conditions first: Negate the statement and simplify (push negation inward) using quantifier laws:
$$\forall x\in\mathbb{R}\ \exists y\in\mathbb{R}\text{ such that } y>x+318.$$
(a) Give the symbolic negation step-by-step.
(b) Translate both the original statement and its negation into plain English t... | [
{
"method_name": "Quantifier Laws",
"approach": "Use $\\neg\\forall=\\exists\\neg$ and $\\neg\\exists=\\forall\\neg$, then negate the inequality.",
"steps": [
"Step 1: Start: $\\neg(\\forall x\\ \\exists y\\ P(x,y))$.",
"Step 2: Apply $\\neg\\forall x\\ Q(x)\\equiv \\exists x\\ \\neg Q(x)$ t... | {
"consistency_check": "Both approaches agree after simplification. Final answer: $\\boxed{\\exists x\\in\\mathbb{R}$.\nThe formal quantifier-negation rules and the English semantics yield the same statement: the existence of an $x$ that upper-bounds all reals after shifting by a constant.",
"robustness_analysis": ... | [
{
"error_description": "Negated only the inequality but kept quantifiers unchanged.",
"why_plausible": "The inequality is the most visible part of the statement, so attention focuses there.",
"why_wrong": "Quantifiers determine scope; the correct negation flips $\\forall\\leftrightarrow\\exists$ and cha... | Key idea: Negating quantified statements requires flipping each quantifier and negating the predicate while keeping variable order. Always sanity-check by translating to plain English. (Here the result is $\boxed{\exists x\in\mathbb{R}$.) |
math-001907 | Foundations: Translating Statements | 1 | Checkpoint: Negate the statement and simplify (push negation inward) using quantifier laws:
$$\forall x\in\mathbb{R}\ \exists y\in\mathbb{R}\text{ such that } y>x+62.$$
(a) Give the symbolic negation step-by-step.
(b) Translate both the original statement and its negation into plain English to confirm the meaning.
In... | [
{
"method_name": "Quantifier Laws",
"approach": "Use $\\neg\\forall=\\exists\\neg$ and $\\neg\\exists=\\forall\\neg$, then negate the inequality.",
"steps": [
"Step 1: Start: $\\neg(\\forall x\\ \\exists y\\ P(x,y))$.",
"Step 2: Apply $\\neg\\forall x\\ Q(x)\\equiv \\exists x\\ \\neg Q(x)$ t... | {
"consistency_check": "Consistency verification shows both paths yield the identical boxed result. Final answer: $\\boxed{\\exists x\\in\\mathbb{R}$.\nThe formal quantifier-negation rules and the English semantics yield the same statement: the existence of an $x$ that upper-bounds all reals after shifting by a const... | [
{
"error_description": "Negated only the inequality but kept quantifiers unchanged.",
"why_plausible": "The inequality is the most visible part of the statement, so attention focuses there.",
"why_wrong": "Quantifiers determine scope; the correct negation flips $\\forall\\leftrightarrow\\exists$ and cha... | Key idea: Negating quantified statements requires flipping each quantifier and negating the predicate while keeping variable order. Always sanity-check by translating to plain English. (Here the result is $\boxed{\exists x\in\mathbb{R}$.) |
math-001908 | Logic: Scope — Order of Quantifiers | 1 | Indicate where a theorem is used: Negate the statement and simplify (push negation inward) using quantifier laws:
$$\forall x\in\mathbb{R}\ \exists y\in\mathbb{R}\text{ such that } y>x+45.$$
(a) Give the symbolic negation step-by-step.
(b) Translate both the original statement and its negation into plain English to co... | [
{
"method_name": "English Semantics Cross-Check",
"approach": "Translate to English, negate carefully, then translate back to symbols.",
"steps": [
"Step 1: Original: 'For every real $x$, there exists a real $y$ that is greater than $x+45$.'",
"Step 2: Negation: 'There exists a real $x$ such... | {
"consistency_check": "Consistency verification shows both paths yield the identical boxed result. Final answer: $\\boxed{\\exists x\\in\\mathbb{R}$.\nThe formal quantifier-negation rules and the English semantics yield the same statement: the existence of an $x$ that upper-bounds all reals after shifting by a const... | [
{
"error_description": "Negated only the inequality but kept quantifiers unchanged.",
"why_plausible": "The inequality is the most visible part of the statement, so attention focuses there.",
"why_wrong": "Quantifiers determine scope; the correct negation flips $\\forall\\leftrightarrow\\exists$ and cha... | Remember: Negating quantified statements requires flipping each quantifier and negating the predicate while keeping variable order. Always sanity-check by translating to plain English. (Here the result is $\boxed{\exists x\in\mathbb{R}$.) |
math-001909 | Logic: Formal vs English Paraphrase | 1 | Explain what is being counted/optimized: Negate the statement and simplify (push negation inward) using quantifier laws:
$$\forall x\in\mathbb{R}\ \exists y\in\mathbb{R}\text{ such that } y>x+313.$$
(a) Give the symbolic negation step-by-step.
(b) Translate both the original statement and its negation into plain Engli... | [
{
"method_name": "Quantifier Laws",
"approach": "Use $\\neg\\forall=\\exists\\neg$ and $\\neg\\exists=\\forall\\neg$, then negate the inequality.",
"steps": [
"Step 1: Start: $\\neg(\\forall x\\ \\exists y\\ P(x,y))$.",
"Step 2: Apply $\\neg\\forall x\\ Q(x)\\equiv \\exists x\\ \\neg Q(x)$ t... | {
"consistency_check": "Both approaches agree after simplification. Final answer: $\\boxed{\\exists x\\in\\mathbb{R}$.\nThe formal quantifier-negation rules and the English semantics yield the same statement: the existence of an $x$ that upper-bounds all reals after shifting by a constant.",
"robustness_analysis": ... | [
{
"error_description": "Negated only the inequality but kept quantifiers unchanged.",
"why_plausible": "The inequality is the most visible part of the statement, so attention focuses there.",
"why_wrong": "Quantifiers determine scope; the correct negation flips $\\forall\\leftrightarrow\\exists$ and cha... | Key idea: Negating quantified statements requires flipping each quantifier and negating the predicate while keeping variable order. Always sanity-check by translating to plain English. |
math-001910 | Logic: Scope — Order of Quantifiers | 1 | Give an answer and a quick verification: Negate the statement and simplify (push negation inward) using quantifier laws:
$$\forall x\in\mathbb{R}\ \exists y\in\mathbb{R}\text{ such that } y>x+182.$$
(a) Give the symbolic negation step-by-step.
(b) Translate both the original statement and its negation into plain Engli... | [
{
"method_name": "English Semantics Cross-Check",
"approach": "Translate to English, negate carefully, then translate back to symbols.",
"steps": [
"Step 1: Original: 'For every real $x$, there exists a real $y$ that is greater than $x+182$.'",
"Step 2: Negation: 'There exists a real $x$ suc... | {
"consistency_check": "Both approaches agree after simplification. Final answer: $\\boxed{\\exists x\\in\\mathbb{R}$.\nThe formal quantifier-negation rules and the English semantics yield the same statement: the existence of an $x$ that upper-bounds all reals after shifting by a constant.",
"robustness_analysis": ... | [
{
"error_description": "Negated only the inequality but kept quantifiers unchanged.",
"why_plausible": "The inequality is the most visible part of the statement, so attention focuses there.",
"why_wrong": "Quantifiers determine scope; the correct negation flips $\\forall\\leftrightarrow\\exists$ and cha... | Core principle: Negating quantified statements requires flipping each quantifier and negating the predicate while keeping variable order. Always sanity-check by translating to plain English. |
math-001911 | Logic: Quantifiers — Negation Rules | 1 | Work this out carefully: Negate the statement and simplify (push negation inward) using quantifier laws:
$$\forall x\in\mathbb{R}\ \exists y\in\mathbb{R}\text{ such that } y>x+109.$$
(a) Give the symbolic negation step-by-step.
(b) Translate both the original statement and its negation into plain English to confirm th... | [
{
"method_name": "English Semantics Cross-Check",
"approach": "Translate to English, negate carefully, then translate back to symbols.",
"steps": [
"Step 1: Original: 'For every real $x$, there exists a real $y$ that is greater than $x+109$.'",
"Step 2: Negation: 'There exists a real $x$ suc... | {
"consistency_check": "Consistency verification shows both paths yield the identical boxed result. Final answer: $\\boxed{\\exists x\\in\\mathbb{R}$.\nThe formal quantifier-negation rules and the English semantics yield the same statement: the existence of an $x$ that upper-bounds all reals after shifting by a const... | [
{
"error_description": "Negated only the inequality but kept quantifiers unchanged.",
"why_plausible": "The inequality is the most visible part of the statement, so attention focuses there.",
"why_wrong": "Quantifiers determine scope; the correct negation flips $\\forall\\leftrightarrow\\exists$ and cha... | Takeaway: Negating quantified statements requires flipping each quantifier and negating the predicate while keeping variable order. Always sanity-check by translating to plain English. |
math-001912 | Foundations: Translating Statements | 1 | Give a fully justified solution: Negate the statement and simplify (push negation inward) using quantifier laws:
$$\forall x\in\mathbb{R}\ \exists y\in\mathbb{R}\text{ such that } y>x+206.$$
(a) Give the symbolic negation step-by-step.
(b) Translate both the original statement and its negation into plain English to co... | [
{
"method_name": "Quantifier Laws",
"approach": "Use $\\neg\\forall=\\exists\\neg$ and $\\neg\\exists=\\forall\\neg$, then negate the inequality.",
"steps": [
"Step 1: Start: $\\neg(\\forall x\\ \\exists y\\ P(x,y))$.",
"Step 2: Apply $\\neg\\forall x\\ Q(x)\\equiv \\exists x\\ \\neg Q(x)$ t... | {
"consistency_check": "Both approaches agree after simplification. Final answer: $\\boxed{\\exists x\\in\\mathbb{R}$.\nThe formal quantifier-negation rules and the English semantics yield the same statement: the existence of an $x$ that upper-bounds all reals after shifting by a constant.",
"robustness_analysis": ... | [
{
"error_description": "Negated only the inequality but kept quantifiers unchanged.",
"why_plausible": "The inequality is the most visible part of the statement, so attention focuses there.",
"why_wrong": "Quantifiers determine scope; the correct negation flips $\\forall\\leftrightarrow\\exists$ and cha... | Takeaway: Negating quantified statements requires flipping each quantifier and negating the predicate while keeping variable order. Always sanity-check by translating to plain English. |
math-001913 | Foundations: Translating Statements | 1 | Provide a rigorous solution: Negate the statement and simplify (push negation inward) using quantifier laws:
$$\forall x\in\mathbb{R}\ \exists y\in\mathbb{R}\text{ such that } y>x+164.$$
(a) Give the symbolic negation step-by-step.
(b) Translate both the original statement and its negation into plain English to confir... | [
{
"method_name": "English Semantics Cross-Check",
"approach": "Translate to English, negate carefully, then translate back to symbols.",
"steps": [
"Step 1: Original: 'For every real $x$, there exists a real $y$ that is greater than $x+164$.'",
"Step 2: Negation: 'There exists a real $x$ suc... | {
"consistency_check": "Both approaches agree after simplification. Final answer: $\\boxed{\\exists x\\in\\mathbb{R}$.\nThe formal quantifier-negation rules and the English semantics yield the same statement: the existence of an $x$ that upper-bounds all reals after shifting by a constant.",
"robustness_analysis": ... | [
{
"error_description": "Negated only the inequality but kept quantifiers unchanged.",
"why_plausible": "The inequality is the most visible part of the statement, so attention focuses there.",
"why_wrong": "Quantifiers determine scope; the correct negation flips $\\forall\\leftrightarrow\\exists$ and cha... | Takeaway: Negating quantified statements requires flipping each quantifier and negating the predicate while keeping variable order. Always sanity-check by translating to plain English. (Here the result is $\boxed{\exists x\in\mathbb{R}$.) |
math-001914 | Logic: Quantifiers — Negation Rules | 1 | Solve and include a self-check: Negate the statement and simplify (push negation inward) using quantifier laws:
$$\forall x\in\mathbb{R}\ \exists y\in\mathbb{R}\text{ such that } y>x+120.$$
(a) Give the symbolic negation step-by-step.
(b) Translate both the original statement and its negation into plain English to con... | [
{
"method_name": "Quantifier Laws",
"approach": "Use $\\neg\\forall=\\exists\\neg$ and $\\neg\\exists=\\forall\\neg$, then negate the inequality.",
"steps": [
"Step 1: Start: $\\neg(\\forall x\\ \\exists y\\ P(x,y))$.",
"Step 2: Apply $\\neg\\forall x\\ Q(x)\\equiv \\exists x\\ \\neg Q(x)$ t... | {
"consistency_check": "Consistency verification shows both paths yield the identical boxed result. Final answer: $\\boxed{\\exists x\\in\\mathbb{R}$.\nThe formal quantifier-negation rules and the English semantics yield the same statement: the existence of an $x$ that upper-bounds all reals after shifting by a const... | [
{
"error_description": "Negated only the inequality but kept quantifiers unchanged.",
"why_plausible": "The inequality is the most visible part of the statement, so attention focuses there.",
"why_wrong": "Quantifiers determine scope; the correct negation flips $\\forall\\leftrightarrow\\exists$ and cha... | Takeaway: Negating quantified statements requires flipping each quantifier and negating the predicate while keeping variable order. Always sanity-check by translating to plain English. (Here the result is $\boxed{\exists x\in\mathbb{R}$.) |
math-001915 | Logic: Scope — Order of Quantifiers | 1 | Track units/moduli carefully: Negate the statement and simplify (push negation inward) using quantifier laws:
$$\forall x\in\mathbb{R}\ \exists y\in\mathbb{R}\text{ such that } y>x+240.$$
(a) Give the symbolic negation step-by-step.
(b) Translate both the original statement and its negation into plain English to confi... | [
{
"method_name": "English Semantics Cross-Check",
"approach": "Translate to English, negate carefully, then translate back to symbols.",
"steps": [
"Step 1: Original: 'For every real $x$, there exists a real $y$ that is greater than $x+240$.'",
"Step 2: Negation: 'There exists a real $x$ suc... | {
"consistency_check": "Both approaches agree after simplification. Final answer: $\\boxed{\\exists x\\in\\mathbb{R}$.\nThe formal quantifier-negation rules and the English semantics yield the same statement: the existence of an $x$ that upper-bounds all reals after shifting by a constant.",
"robustness_analysis": ... | [
{
"error_description": "Negated only the inequality but kept quantifiers unchanged.",
"why_plausible": "The inequality is the most visible part of the statement, so attention focuses there.",
"why_wrong": "Quantifiers determine scope; the correct negation flips $\\forall\\leftrightarrow\\exists$ and cha... | Remember: Negating quantified statements requires flipping each quantifier and negating the predicate while keeping variable order. Always sanity-check by translating to plain English. (Here the result is $\boxed{\exists x\in\mathbb{R}$.) |
math-001916 | Logic: Quantifiers — Negation Rules | 1 | Find the exact value: Negate the statement and simplify (push negation inward) using quantifier laws:
$$\forall x\in\mathbb{R}\ \exists y\in\mathbb{R}\text{ such that } y>x+380.$$
(a) Give the symbolic negation step-by-step.
(b) Translate both the original statement and its negation into plain English to confirm the m... | [
{
"method_name": "Quantifier Laws",
"approach": "Use $\\neg\\forall=\\exists\\neg$ and $\\neg\\exists=\\forall\\neg$, then negate the inequality.",
"steps": [
"Step 1: Start: $\\neg(\\forall x\\ \\exists y\\ P(x,y))$.",
"Step 2: Apply $\\neg\\forall x\\ Q(x)\\equiv \\exists x\\ \\neg Q(x)$ t... | {
"consistency_check": "Consistency verification shows both paths yield the identical boxed result. Final answer: $\\boxed{\\exists x\\in\\mathbb{R}$.\nThe formal quantifier-negation rules and the English semantics yield the same statement: the existence of an $x$ that upper-bounds all reals after shifting by a const... | [
{
"error_description": "Negated only the inequality but kept quantifiers unchanged.",
"why_plausible": "The inequality is the most visible part of the statement, so attention focuses there.",
"why_wrong": "Quantifiers determine scope; the correct negation flips $\\forall\\leftrightarrow\\exists$ and cha... | Key idea: Negating quantified statements requires flipping each quantifier and negating the predicate while keeping variable order. Always sanity-check by translating to plain English. (Here the result is $\boxed{\exists x\in\mathbb{R}$.) |
math-001917 | Foundations: Translating Statements | 1 | Show all reasoning: Negate the statement and simplify (push negation inward) using quantifier laws:
$$\forall x\in\mathbb{R}\ \exists y\in\mathbb{R}\text{ such that } y>x+371.$$
(a) Give the symbolic negation step-by-step.
(b) Translate both the original statement and its negation into plain English to confirm the mea... | [
{
"method_name": "Quantifier Laws",
"approach": "Use $\\neg\\forall=\\exists\\neg$ and $\\neg\\exists=\\forall\\neg$, then negate the inequality.",
"steps": [
"Step 1: Start: $\\neg(\\forall x\\ \\exists y\\ P(x,y))$.",
"Step 2: Apply $\\neg\\forall x\\ Q(x)\\equiv \\exists x\\ \\neg Q(x)$ t... | {
"consistency_check": "Cross-check: both derivations land on the same invariant quantity. Final answer: $\\boxed{\\exists x\\in\\mathbb{R}$.\nThe formal quantifier-negation rules and the English semantics yield the same statement: the existence of an $x$ that upper-bounds all reals after shifting by a constant.",
... | [
{
"error_description": "Negated only the inequality but kept quantifiers unchanged.",
"why_plausible": "The inequality is the most visible part of the statement, so attention focuses there.",
"why_wrong": "Quantifiers determine scope; the correct negation flips $\\forall\\leftrightarrow\\exists$ and cha... | Core principle: Negating quantified statements requires flipping each quantifier and negating the predicate while keeping variable order. Always sanity-check by translating to plain English. |
math-001918 | Logic: Quantifiers — Negation Rules | 1 | Complete the analysis: Negate the statement and simplify (push negation inward) using quantifier laws:
$$\forall x\in\mathbb{R}\ \exists y\in\mathbb{R}\text{ such that } y>x+99.$$
(a) Give the symbolic negation step-by-step.
(b) Translate both the original statement and its negation into plain English to confirm the m... | [
{
"method_name": "English Semantics Cross-Check",
"approach": "Translate to English, negate carefully, then translate back to symbols.",
"steps": [
"Step 1: Original: 'For every real $x$, there exists a real $y$ that is greater than $x+99$.'",
"Step 2: Negation: 'There exists a real $x$ such... | {
"consistency_check": "The two methods are consistent and must coincide. Final answer: $\\boxed{\\exists x\\in\\mathbb{R}$.\nThe formal quantifier-negation rules and the English semantics yield the same statement: the existence of an $x$ that upper-bounds all reals after shifting by a constant.",
"robustness_analy... | [
{
"error_description": "Negated only the inequality but kept quantifiers unchanged.",
"why_plausible": "The inequality is the most visible part of the statement, so attention focuses there.",
"why_wrong": "Quantifiers determine scope; the correct negation flips $\\forall\\leftrightarrow\\exists$ and cha... | Key idea: Negating quantified statements requires flipping each quantifier and negating the predicate while keeping variable order. Always sanity-check by translating to plain English. (Here the result is $\boxed{\exists x\in\mathbb{R}$.) |
math-001919 | Logic: Quantifiers — Negation Rules | 1 | Answer using clear logical steps: Negate the statement and simplify (push negation inward) using quantifier laws:
$$\forall x\in\mathbb{R}\ \exists y\in\mathbb{R}\text{ such that } y>x+172.$$
(a) Give the symbolic negation step-by-step.
(b) Translate both the original statement and its negation into plain English to c... | [
{
"method_name": "English Semantics Cross-Check",
"approach": "Translate to English, negate carefully, then translate back to symbols.",
"steps": [
"Step 1: Original: 'For every real $x$, there exists a real $y$ that is greater than $x+172$.'",
"Step 2: Negation: 'There exists a real $x$ suc... | {
"consistency_check": "Cross-check: both derivations land on the same invariant quantity. Final answer: $\\boxed{\\exists x\\in\\mathbb{R}$.\nThe formal quantifier-negation rules and the English semantics yield the same statement: the existence of an $x$ that upper-bounds all reals after shifting by a constant.",
... | [
{
"error_description": "Negated only the inequality but kept quantifiers unchanged.",
"why_plausible": "The inequality is the most visible part of the statement, so attention focuses there.",
"why_wrong": "Quantifiers determine scope; the correct negation flips $\\forall\\leftrightarrow\\exists$ and cha... | Remember: Negating quantified statements requires flipping each quantifier and negating the predicate while keeping variable order. Always sanity-check by translating to plain English. (Here the result is $\boxed{\exists x\in\mathbb{R}$.) |
math-001920 | Logic: Scope — Order of Quantifiers | 1 | Be explicit about assumptions: Negate the statement and simplify (push negation inward) using quantifier laws:
$$\forall x\in\mathbb{R}\ \exists y\in\mathbb{R}\text{ such that } y>x+127.$$
(a) Give the symbolic negation step-by-step.
(b) Translate both the original statement and its negation into plain English to conf... | [
{
"method_name": "English Semantics Cross-Check",
"approach": "Translate to English, negate carefully, then translate back to symbols.",
"steps": [
"Step 1: Original: 'For every real $x$, there exists a real $y$ that is greater than $x+127$.'",
"Step 2: Negation: 'There exists a real $x$ suc... | {
"consistency_check": "Consistency verification shows both paths yield the identical boxed result. Final answer: $\\boxed{\\exists x\\in\\mathbb{R}$.\nThe formal quantifier-negation rules and the English semantics yield the same statement: the existence of an $x$ that upper-bounds all reals after shifting by a const... | [
{
"error_description": "Negated only the inequality but kept quantifiers unchanged.",
"why_plausible": "The inequality is the most visible part of the statement, so attention focuses there.",
"why_wrong": "Quantifiers determine scope; the correct negation flips $\\forall\\leftrightarrow\\exists$ and cha... | Takeaway: Negating quantified statements requires flipping each quantifier and negating the predicate while keeping variable order. Always sanity-check by translating to plain English. (Here the result is $\boxed{\exists x\in\mathbb{R}$.) |
math-001921 | Foundations: Translating Statements | 1 | Track quantifiers carefully: Negate the statement and simplify (push negation inward) using quantifier laws:
$$\forall x\in\mathbb{R}\ \exists y\in\mathbb{R}\text{ such that } y>x+92.$$
(a) Give the symbolic negation step-by-step.
(b) Translate both the original statement and its negation into plain English to confirm... | [
{
"method_name": "English Semantics Cross-Check",
"approach": "Translate to English, negate carefully, then translate back to symbols.",
"steps": [
"Step 1: Original: 'For every real $x$, there exists a real $y$ that is greater than $x+92$.'",
"Step 2: Negation: 'There exists a real $x$ such... | {
"consistency_check": "Consistency verification shows both paths yield the identical boxed result. Final answer: $\\boxed{\\exists x\\in\\mathbb{R}$.\nThe formal quantifier-negation rules and the English semantics yield the same statement: the existence of an $x$ that upper-bounds all reals after shifting by a const... | [
{
"error_description": "Negated only the inequality but kept quantifiers unchanged.",
"why_plausible": "The inequality is the most visible part of the statement, so attention focuses there.",
"why_wrong": "Quantifiers determine scope; the correct negation flips $\\forall\\leftrightarrow\\exists$ and cha... | Takeaway: Negating quantified statements requires flipping each quantifier and negating the predicate while keeping variable order. Always sanity-check by translating to plain English. (Here the result is $\boxed{\exists x\in\mathbb{R}$.) |
math-001922 | Logic: Scope — Order of Quantifiers | 1 | Explain each transformation: Negate the statement and simplify (push negation inward) using quantifier laws:
$$\forall x\in\mathbb{R}\ \exists y\in\mathbb{R}\text{ such that } y>x+290.$$
(a) Give the symbolic negation step-by-step.
(b) Translate both the original statement and its negation into plain English to confir... | [
{
"method_name": "Quantifier Laws",
"approach": "Use $\\neg\\forall=\\exists\\neg$ and $\\neg\\exists=\\forall\\neg$, then negate the inequality.",
"steps": [
"Step 1: Start: $\\neg(\\forall x\\ \\exists y\\ P(x,y))$.",
"Step 2: Apply $\\neg\\forall x\\ Q(x)\\equiv \\exists x\\ \\neg Q(x)$ t... | {
"consistency_check": "Cross-check: both derivations land on the same invariant quantity. Final answer: $\\boxed{\\exists x\\in\\mathbb{R}$.\nThe formal quantifier-negation rules and the English semantics yield the same statement: the existence of an $x$ that upper-bounds all reals after shifting by a constant.",
... | [
{
"error_description": "Negated only the inequality but kept quantifiers unchanged.",
"why_plausible": "The inequality is the most visible part of the statement, so attention focuses there.",
"why_wrong": "Quantifiers determine scope; the correct negation flips $\\forall\\leftrightarrow\\exists$ and cha... | Key idea: Negating quantified statements requires flipping each quantifier and negating the predicate while keeping variable order. Always sanity-check by translating to plain English. |
math-001923 | Logic: Formal vs English Paraphrase | 1 | Make each step logically reversible (or explain if not): Negate the statement and simplify (push negation inward) using quantifier laws:
$$\forall x\in\mathbb{R}\ \exists y\in\mathbb{R}\text{ such that } y>x+171.$$
(a) Give the symbolic negation step-by-step.
(b) Translate both the original statement and its negation ... | [
{
"method_name": "English Semantics Cross-Check",
"approach": "Translate to English, negate carefully, then translate back to symbols.",
"steps": [
"Step 1: Original: 'For every real $x$, there exists a real $y$ that is greater than $x+171$.'",
"Step 2: Negation: 'There exists a real $x$ suc... | {
"consistency_check": "Consistency verification shows both paths yield the identical boxed result. Final answer: $\\boxed{\\exists x\\in\\mathbb{R}$.\nThe formal quantifier-negation rules and the English semantics yield the same statement: the existence of an $x$ that upper-bounds all reals after shifting by a const... | [
{
"error_description": "Negated only the inequality but kept quantifiers unchanged.",
"why_plausible": "The inequality is the most visible part of the statement, so attention focuses there.",
"why_wrong": "Quantifiers determine scope; the correct negation flips $\\forall\\leftrightarrow\\exists$ and cha... | Key idea: Negating quantified statements requires flipping each quantifier and negating the predicate while keeping variable order. Always sanity-check by translating to plain English. |
math-001924 | Logic: Quantifiers — Negation Rules | 1 | Solve and sanity-check: Negate the statement and simplify (push negation inward) using quantifier laws:
$$\forall x\in\mathbb{R}\ \exists y\in\mathbb{R}\text{ such that } y>x+335.$$
(a) Give the symbolic negation step-by-step.
(b) Translate both the original statement and its negation into plain English to confirm the... | [
{
"method_name": "English Semantics Cross-Check",
"approach": "Translate to English, negate carefully, then translate back to symbols.",
"steps": [
"Step 1: Original: 'For every real $x$, there exists a real $y$ that is greater than $x+335$.'",
"Step 2: Negation: 'There exists a real $x$ suc... | {
"consistency_check": "Consistency verification shows both paths yield the identical boxed result. Final answer: $\\boxed{\\exists x\\in\\mathbb{R}$.\nThe formal quantifier-negation rules and the English semantics yield the same statement: the existence of an $x$ that upper-bounds all reals after shifting by a const... | [
{
"error_description": "Negated only the inequality but kept quantifiers unchanged.",
"why_plausible": "The inequality is the most visible part of the statement, so attention focuses there.",
"why_wrong": "Quantifiers determine scope; the correct negation flips $\\forall\\leftrightarrow\\exists$ and cha... | Core principle: Negating quantified statements requires flipping each quantifier and negating the predicate while keeping variable order. Always sanity-check by translating to plain English. (Here the result is $\boxed{\exists x\in\mathbb{R}$.) |
math-001925 | Logic: Formal vs English Paraphrase | 1 | Make each step logically reversible (or explain if not): Negate the statement and simplify (push negation inward) using quantifier laws:
$$\forall x\in\mathbb{R}\ \exists y\in\mathbb{R}\text{ such that } y>x+205.$$
(a) Give the symbolic negation step-by-step.
(b) Translate both the original statement and its negation ... | [
{
"method_name": "Quantifier Laws",
"approach": "Use $\\neg\\forall=\\exists\\neg$ and $\\neg\\exists=\\forall\\neg$, then negate the inequality.",
"steps": [
"Step 1: Start: $\\neg(\\forall x\\ \\exists y\\ P(x,y))$.",
"Step 2: Apply $\\neg\\forall x\\ Q(x)\\equiv \\exists x\\ \\neg Q(x)$ t... | {
"consistency_check": "Cross-check: both derivations land on the same invariant quantity. Final answer: $\\boxed{\\exists x\\in\\mathbb{R}$.\nThe formal quantifier-negation rules and the English semantics yield the same statement: the existence of an $x$ that upper-bounds all reals after shifting by a constant.",
... | [
{
"error_description": "Negated only the inequality but kept quantifiers unchanged.",
"why_plausible": "The inequality is the most visible part of the statement, so attention focuses there.",
"why_wrong": "Quantifiers determine scope; the correct negation flips $\\forall\\leftrightarrow\\exists$ and cha... | Core principle: Negating quantified statements requires flipping each quantifier and negating the predicate while keeping variable order. Always sanity-check by translating to plain English. (Here the result is $\boxed{\exists x\in\mathbb{R}$.) |
math-001926 | Logic: Scope — Order of Quantifiers | 1 | Warm-up: Negate the statement and simplify (push negation inward) using quantifier laws:
$$\forall x\in\mathbb{R}\ \exists y\in\mathbb{R}\text{ such that } y>x+55.$$
(a) Give the symbolic negation step-by-step.
(b) Translate both the original statement and its negation into plain English to confirm the meaning.
Inclu... | [
{
"method_name": "Quantifier Laws",
"approach": "Use $\\neg\\forall=\\exists\\neg$ and $\\neg\\exists=\\forall\\neg$, then negate the inequality.",
"steps": [
"Step 1: Start: $\\neg(\\forall x\\ \\exists y\\ P(x,y))$.",
"Step 2: Apply $\\neg\\forall x\\ Q(x)\\equiv \\exists x\\ \\neg Q(x)$ t... | {
"consistency_check": "Consistency verification shows both paths yield the identical boxed result. Final answer: $\\boxed{\\exists x\\in\\mathbb{R}$.\nThe formal quantifier-negation rules and the English semantics yield the same statement: the existence of an $x$ that upper-bounds all reals after shifting by a const... | [
{
"error_description": "Negated only the inequality but kept quantifiers unchanged.",
"why_plausible": "The inequality is the most visible part of the statement, so attention focuses there.",
"why_wrong": "Quantifiers determine scope; the correct negation flips $\\forall\\leftrightarrow\\exists$ and cha... | Remember: Negating quantified statements requires flipping each quantifier and negating the predicate while keeping variable order. Always sanity-check by translating to plain English. |
math-001927 | Foundations: Translating Statements | 1 | Keep the final answer in boxed form: Negate the statement and simplify (push negation inward) using quantifier laws:
$$\forall x\in\mathbb{R}\ \exists y\in\mathbb{R}\text{ such that } y>x+259.$$
(a) Give the symbolic negation step-by-step.
(b) Translate both the original statement and its negation into plain English t... | [
{
"method_name": "English Semantics Cross-Check",
"approach": "Translate to English, negate carefully, then translate back to symbols.",
"steps": [
"Step 1: Original: 'For every real $x$, there exists a real $y$ that is greater than $x+259$.'",
"Step 2: Negation: 'There exists a real $x$ suc... | {
"consistency_check": "The two methods are consistent and must coincide. Final answer: $\\boxed{\\exists x\\in\\mathbb{R}$.\nThe formal quantifier-negation rules and the English semantics yield the same statement: the existence of an $x$ that upper-bounds all reals after shifting by a constant.",
"robustness_analy... | [
{
"error_description": "Negated only the inequality but kept quantifiers unchanged.",
"why_plausible": "The inequality is the most visible part of the statement, so attention focuses there.",
"why_wrong": "Quantifiers determine scope; the correct negation flips $\\forall\\leftrightarrow\\exists$ and cha... | Takeaway: Negating quantified statements requires flipping each quantifier and negating the predicate while keeping variable order. Always sanity-check by translating to plain English. (Here the result is $\boxed{\exists x\in\mathbb{R}$.) |
math-001928 | Logic: Formal vs English Paraphrase | 1 | Derive the result step-by-step: Negate the statement and simplify (push negation inward) using quantifier laws:
$$\forall x\in\mathbb{R}\ \exists y\in\mathbb{R}\text{ such that } y>x+151.$$
(a) Give the symbolic negation step-by-step.
(b) Translate both the original statement and its negation into plain English to con... | [
{
"method_name": "English Semantics Cross-Check",
"approach": "Translate to English, negate carefully, then translate back to symbols.",
"steps": [
"Step 1: Original: 'For every real $x$, there exists a real $y$ that is greater than $x+151$.'",
"Step 2: Negation: 'There exists a real $x$ suc... | {
"consistency_check": "Cross-check: both derivations land on the same invariant quantity. Final answer: $\\boxed{\\exists x\\in\\mathbb{R}$.\nThe formal quantifier-negation rules and the English semantics yield the same statement: the existence of an $x$ that upper-bounds all reals after shifting by a constant.",
... | [
{
"error_description": "Negated only the inequality but kept quantifiers unchanged.",
"why_plausible": "The inequality is the most visible part of the statement, so attention focuses there.",
"why_wrong": "Quantifiers determine scope; the correct negation flips $\\forall\\leftrightarrow\\exists$ and cha... | Key idea: Negating quantified statements requires flipping each quantifier and negating the predicate while keeping variable order. Always sanity-check by translating to plain English. |
math-001929 | Logic: Scope — Order of Quantifiers | 1 | Solve and include a self-check: Negate the statement and simplify (push negation inward) using quantifier laws:
$$\forall x\in\mathbb{R}\ \exists y\in\mathbb{R}\text{ such that } y>x+42.$$
(a) Give the symbolic negation step-by-step.
(b) Translate both the original statement and its negation into plain English to conf... | [
{
"method_name": "English Semantics Cross-Check",
"approach": "Translate to English, negate carefully, then translate back to symbols.",
"steps": [
"Step 1: Original: 'For every real $x$, there exists a real $y$ that is greater than $x+42$.'",
"Step 2: Negation: 'There exists a real $x$ such... | {
"consistency_check": "Both approaches agree after simplification. Final answer: $\\boxed{\\exists x\\in\\mathbb{R}$.\nThe formal quantifier-negation rules and the English semantics yield the same statement: the existence of an $x$ that upper-bounds all reals after shifting by a constant.",
"robustness_analysis": ... | [
{
"error_description": "Negated only the inequality but kept quantifiers unchanged.",
"why_plausible": "The inequality is the most visible part of the statement, so attention focuses there.",
"why_wrong": "Quantifiers determine scope; the correct negation flips $\\forall\\leftrightarrow\\exists$ and cha... | Remember: Negating quantified statements requires flipping each quantifier and negating the predicate while keeping variable order. Always sanity-check by translating to plain English. |
math-001930 | Foundations: Translating Statements | 1 | Where appropriate, name the theorem you use: Negate the statement and simplify (push negation inward) using quantifier laws:
$$\forall x\in\mathbb{R}\ \exists y\in\mathbb{R}\text{ such that } y>x+340.$$
(a) Give the symbolic negation step-by-step.
(b) Translate both the original statement and its negation into plain E... | [
{
"method_name": "Quantifier Laws",
"approach": "Use $\\neg\\forall=\\exists\\neg$ and $\\neg\\exists=\\forall\\neg$, then negate the inequality.",
"steps": [
"Step 1: Start: $\\neg(\\forall x\\ \\exists y\\ P(x,y))$.",
"Step 2: Apply $\\neg\\forall x\\ Q(x)\\equiv \\exists x\\ \\neg Q(x)$ t... | {
"consistency_check": "Consistency verification shows both paths yield the identical boxed result. Final answer: $\\boxed{\\exists x\\in\\mathbb{R}$.\nThe formal quantifier-negation rules and the English semantics yield the same statement: the existence of an $x$ that upper-bounds all reals after shifting by a const... | [
{
"error_description": "Negated only the inequality but kept quantifiers unchanged.",
"why_plausible": "The inequality is the most visible part of the statement, so attention focuses there.",
"why_wrong": "Quantifiers determine scope; the correct negation flips $\\forall\\leftrightarrow\\exists$ and cha... | Takeaway: Negating quantified statements requires flipping each quantifier and negating the predicate while keeping variable order. Always sanity-check by translating to plain English. (Here the result is $\boxed{\exists x\in\mathbb{R}$.) |
math-001931 | Logic: Quantifiers — Negation Rules | 1 | Solve with verification: Negate the statement and simplify (push negation inward) using quantifier laws:
$$\forall x\in\mathbb{R}\ \exists y\in\mathbb{R}\text{ such that } y>x+167.$$
(a) Give the symbolic negation step-by-step.
(b) Translate both the original statement and its negation into plain English to confirm th... | [
{
"method_name": "English Semantics Cross-Check",
"approach": "Translate to English, negate carefully, then translate back to symbols.",
"steps": [
"Step 1: Original: 'For every real $x$, there exists a real $y$ that is greater than $x+167$.'",
"Step 2: Negation: 'There exists a real $x$ suc... | {
"consistency_check": "The two methods are consistent and must coincide. Final answer: $\\boxed{\\exists x\\in\\mathbb{R}$.\nThe formal quantifier-negation rules and the English semantics yield the same statement: the existence of an $x$ that upper-bounds all reals after shifting by a constant.",
"robustness_analy... | [
{
"error_description": "Negated only the inequality but kept quantifiers unchanged.",
"why_plausible": "The inequality is the most visible part of the statement, so attention focuses there.",
"why_wrong": "Quantifiers determine scope; the correct negation flips $\\forall\\leftrightarrow\\exists$ and cha... | Remember: Negating quantified statements requires flipping each quantifier and negating the predicate while keeping variable order. Always sanity-check by translating to plain English. (Here the result is $\boxed{\exists x\in\mathbb{R}$.) |
math-001932 | Logic: Scope — Order of Quantifiers | 1 | Solve and include a self-check: Negate the statement and simplify (push negation inward) using quantifier laws:
$$\forall x\in\mathbb{R}\ \exists y\in\mathbb{R}\text{ such that } y>x+324.$$
(a) Give the symbolic negation step-by-step.
(b) Translate both the original statement and its negation into plain English to con... | [
{
"method_name": "English Semantics Cross-Check",
"approach": "Translate to English, negate carefully, then translate back to symbols.",
"steps": [
"Step 1: Original: 'For every real $x$, there exists a real $y$ that is greater than $x+324$.'",
"Step 2: Negation: 'There exists a real $x$ suc... | {
"consistency_check": "Both approaches agree after simplification. Final answer: $\\boxed{\\exists x\\in\\mathbb{R}$.\nThe formal quantifier-negation rules and the English semantics yield the same statement: the existence of an $x$ that upper-bounds all reals after shifting by a constant.",
"robustness_analysis": ... | [
{
"error_description": "Negated only the inequality but kept quantifiers unchanged.",
"why_plausible": "The inequality is the most visible part of the statement, so attention focuses there.",
"why_wrong": "Quantifiers determine scope; the correct negation flips $\\forall\\leftrightarrow\\exists$ and cha... | Core principle: Negating quantified statements requires flipping each quantifier and negating the predicate while keeping variable order. Always sanity-check by translating to plain English. (Here the result is $\boxed{\exists x\in\mathbb{R}$.) |
math-001933 | Logic: Scope — Order of Quantifiers | 1 | Find the exact value: Negate the statement and simplify (push negation inward) using quantifier laws:
$$\forall x\in\mathbb{R}\ \exists y\in\mathbb{R}\text{ such that } y>x+374.$$
(a) Give the symbolic negation step-by-step.
(b) Translate both the original statement and its negation into plain English to confirm the m... | [
{
"method_name": "English Semantics Cross-Check",
"approach": "Translate to English, negate carefully, then translate back to symbols.",
"steps": [
"Step 1: Original: 'For every real $x$, there exists a real $y$ that is greater than $x+374$.'",
"Step 2: Negation: 'There exists a real $x$ suc... | {
"consistency_check": "Consistency verification shows both paths yield the identical boxed result. Final answer: $\\boxed{\\exists x\\in\\mathbb{R}$.\nThe formal quantifier-negation rules and the English semantics yield the same statement: the existence of an $x$ that upper-bounds all reals after shifting by a const... | [
{
"error_description": "Negated only the inequality but kept quantifiers unchanged.",
"why_plausible": "The inequality is the most visible part of the statement, so attention focuses there.",
"why_wrong": "Quantifiers determine scope; the correct negation flips $\\forall\\leftrightarrow\\exists$ and cha... | Core principle: Negating quantified statements requires flipping each quantifier and negating the predicate while keeping variable order. Always sanity-check by translating to plain English. |
math-001934 | Logic: Formal vs English Paraphrase | 1 | Warm-up: Negate the statement and simplify (push negation inward) using quantifier laws:
$$\forall x\in\mathbb{R}\ \exists y\in\mathbb{R}\text{ such that } y>x+357.$$
(a) Give the symbolic negation step-by-step.
(b) Translate both the original statement and its negation into plain English to confirm the meaning.
Incl... | [
{
"method_name": "Quantifier Laws",
"approach": "Use $\\neg\\forall=\\exists\\neg$ and $\\neg\\exists=\\forall\\neg$, then negate the inequality.",
"steps": [
"Step 1: Start: $\\neg(\\forall x\\ \\exists y\\ P(x,y))$.",
"Step 2: Apply $\\neg\\forall x\\ Q(x)\\equiv \\exists x\\ \\neg Q(x)$ t... | {
"consistency_check": "Consistency verification shows both paths yield the identical boxed result. Final answer: $\\boxed{\\exists x\\in\\mathbb{R}$.\nThe formal quantifier-negation rules and the English semantics yield the same statement: the existence of an $x$ that upper-bounds all reals after shifting by a const... | [
{
"error_description": "Negated only the inequality but kept quantifiers unchanged.",
"why_plausible": "The inequality is the most visible part of the statement, so attention focuses there.",
"why_wrong": "Quantifiers determine scope; the correct negation flips $\\forall\\leftrightarrow\\exists$ and cha... | Core principle: Negating quantified statements requires flipping each quantifier and negating the predicate while keeping variable order. Always sanity-check by translating to plain English. (Here the result is $\boxed{\exists x\in\mathbb{R}$.) |
math-001935 | Foundations: Translating Statements | 1 | Make each step logically reversible (or explain if not): Negate the statement and simplify (push negation inward) using quantifier laws:
$$\forall x\in\mathbb{R}\ \exists y\in\mathbb{R}\text{ such that } y>x+299.$$
(a) Give the symbolic negation step-by-step.
(b) Translate both the original statement and its negation ... | [
{
"method_name": "English Semantics Cross-Check",
"approach": "Translate to English, negate carefully, then translate back to symbols.",
"steps": [
"Step 1: Original: 'For every real $x$, there exists a real $y$ that is greater than $x+299$.'",
"Step 2: Negation: 'There exists a real $x$ suc... | {
"consistency_check": "Consistency verification shows both paths yield the identical boxed result. Final answer: $\\boxed{\\exists x\\in\\mathbb{R}$.\nThe formal quantifier-negation rules and the English semantics yield the same statement: the existence of an $x$ that upper-bounds all reals after shifting by a const... | [
{
"error_description": "Negated only the inequality but kept quantifiers unchanged.",
"why_plausible": "The inequality is the most visible part of the statement, so attention focuses there.",
"why_wrong": "Quantifiers determine scope; the correct negation flips $\\forall\\leftrightarrow\\exists$ and cha... | Takeaway: Negating quantified statements requires flipping each quantifier and negating the predicate while keeping variable order. Always sanity-check by translating to plain English. (Here the result is $\boxed{\exists x\in\mathbb{R}$.) |
math-001936 | Logic: Quantifiers — Negation Rules | 1 | Try to avoid pattern-matching; explain why: Negate the statement and simplify (push negation inward) using quantifier laws:
$$\forall x\in\mathbb{R}\ \exists y\in\mathbb{R}\text{ such that } y>x+76.$$
(a) Give the symbolic negation step-by-step.
(b) Translate both the original statement and its negation into plain Eng... | [
{
"method_name": "Quantifier Laws",
"approach": "Use $\\neg\\forall=\\exists\\neg$ and $\\neg\\exists=\\forall\\neg$, then negate the inequality.",
"steps": [
"Step 1: Start: $\\neg(\\forall x\\ \\exists y\\ P(x,y))$.",
"Step 2: Apply $\\neg\\forall x\\ Q(x)\\equiv \\exists x\\ \\neg Q(x)$ t... | {
"consistency_check": "Both approaches agree after simplification. Final answer: $\\boxed{\\exists x\\in\\mathbb{R}$.\nThe formal quantifier-negation rules and the English semantics yield the same statement: the existence of an $x$ that upper-bounds all reals after shifting by a constant.",
"robustness_analysis": ... | [
{
"error_description": "Negated only the inequality but kept quantifiers unchanged.",
"why_plausible": "The inequality is the most visible part of the statement, so attention focuses there.",
"why_wrong": "Quantifiers determine scope; the correct negation flips $\\forall\\leftrightarrow\\exists$ and cha... | Core principle: Negating quantified statements requires flipping each quantifier and negating the predicate while keeping variable order. Always sanity-check by translating to plain English. (Here the result is $\boxed{\exists x\in\mathbb{R}$.) |
math-001937 | Logic: Scope — Order of Quantifiers | 1 | Answer using clear logical steps: Negate the statement and simplify (push negation inward) using quantifier laws:
$$\forall x\in\mathbb{R}\ \exists y\in\mathbb{R}\text{ such that } y>x+355.$$
(a) Give the symbolic negation step-by-step.
(b) Translate both the original statement and its negation into plain English to c... | [
{
"method_name": "English Semantics Cross-Check",
"approach": "Translate to English, negate carefully, then translate back to symbols.",
"steps": [
"Step 1: Original: 'For every real $x$, there exists a real $y$ that is greater than $x+355$.'",
"Step 2: Negation: 'There exists a real $x$ suc... | {
"consistency_check": "Cross-check: both derivations land on the same invariant quantity. Final answer: $\\boxed{\\exists x\\in\\mathbb{R}$.\nThe formal quantifier-negation rules and the English semantics yield the same statement: the existence of an $x$ that upper-bounds all reals after shifting by a constant.",
... | [
{
"error_description": "Negated only the inequality but kept quantifiers unchanged.",
"why_plausible": "The inequality is the most visible part of the statement, so attention focuses there.",
"why_wrong": "Quantifiers determine scope; the correct negation flips $\\forall\\leftrightarrow\\exists$ and cha... | Core principle: Negating quantified statements requires flipping each quantifier and negating the predicate while keeping variable order. Always sanity-check by translating to plain English. |
math-001938 | Logic: Formal vs English Paraphrase | 1 | Give a theorem-based solution: Negate the statement and simplify (push negation inward) using quantifier laws:
$$\forall x\in\mathbb{R}\ \exists y\in\mathbb{R}\text{ such that } y>x+378.$$
(a) Give the symbolic negation step-by-step.
(b) Translate both the original statement and its negation into plain English to conf... | [
{
"method_name": "English Semantics Cross-Check",
"approach": "Translate to English, negate carefully, then translate back to symbols.",
"steps": [
"Step 1: Original: 'For every real $x$, there exists a real $y$ that is greater than $x+378$.'",
"Step 2: Negation: 'There exists a real $x$ suc... | {
"consistency_check": "The two methods are consistent and must coincide. Final answer: $\\boxed{\\exists x\\in\\mathbb{R}$.\nThe formal quantifier-negation rules and the English semantics yield the same statement: the existence of an $x$ that upper-bounds all reals after shifting by a constant.",
"robustness_analy... | [
{
"error_description": "Negated only the inequality but kept quantifiers unchanged.",
"why_plausible": "The inequality is the most visible part of the statement, so attention focuses there.",
"why_wrong": "Quantifiers determine scope; the correct negation flips $\\forall\\leftrightarrow\\exists$ and cha... | Takeaway: Negating quantified statements requires flipping each quantifier and negating the predicate while keeping variable order. Always sanity-check by translating to plain English. |
math-001939 | Logic: Scope — Order of Quantifiers | 1 | Solve and include a self-check: Negate the statement and simplify (push negation inward) using quantifier laws:
$$\forall x\in\mathbb{R}\ \exists y\in\mathbb{R}\text{ such that } y>x+154.$$
(a) Give the symbolic negation step-by-step.
(b) Translate both the original statement and its negation into plain English to con... | [
{
"method_name": "English Semantics Cross-Check",
"approach": "Translate to English, negate carefully, then translate back to symbols.",
"steps": [
"Step 1: Original: 'For every real $x$, there exists a real $y$ that is greater than $x+154$.'",
"Step 2: Negation: 'There exists a real $x$ suc... | {
"consistency_check": "Both approaches agree after simplification. Final answer: $\\boxed{\\exists x\\in\\mathbb{R}$.\nThe formal quantifier-negation rules and the English semantics yield the same statement: the existence of an $x$ that upper-bounds all reals after shifting by a constant.",
"robustness_analysis": ... | [
{
"error_description": "Negated only the inequality but kept quantifiers unchanged.",
"why_plausible": "The inequality is the most visible part of the statement, so attention focuses there.",
"why_wrong": "Quantifiers determine scope; the correct negation flips $\\forall\\leftrightarrow\\exists$ and cha... | Takeaway: Negating quantified statements requires flipping each quantifier and negating the predicate while keeping variable order. Always sanity-check by translating to plain English. (Here the result is $\boxed{\exists x\in\mathbb{R}$.) |
math-001940 | Logic: Scope — Order of Quantifiers | 1 | Question: Negate the statement and simplify (push negation inward) using quantifier laws:
$$\forall x\in\mathbb{R}\ \exists y\in\mathbb{R}\text{ such that } y>x+68.$$
(a) Give the symbolic negation step-by-step.
(b) Translate both the original statement and its negation into plain English to confirm the meaning.
Incl... | [
{
"method_name": "English Semantics Cross-Check",
"approach": "Translate to English, negate carefully, then translate back to symbols.",
"steps": [
"Step 1: Original: 'For every real $x$, there exists a real $y$ that is greater than $x+68$.'",
"Step 2: Negation: 'There exists a real $x$ such... | {
"consistency_check": "The two methods are consistent and must coincide. Final answer: $\\boxed{\\exists x\\in\\mathbb{R}$.\nThe formal quantifier-negation rules and the English semantics yield the same statement: the existence of an $x$ that upper-bounds all reals after shifting by a constant.",
"robustness_analy... | [
{
"error_description": "Negated only the inequality but kept quantifiers unchanged.",
"why_plausible": "The inequality is the most visible part of the statement, so attention focuses there.",
"why_wrong": "Quantifiers determine scope; the correct negation flips $\\forall\\leftrightarrow\\exists$ and cha... | Takeaway: Negating quantified statements requires flipping each quantifier and negating the predicate while keeping variable order. Always sanity-check by translating to plain English. |
math-001941 | Foundations: Translating Statements | 1 | Explain why your operations are valid: Negate the statement and simplify (push negation inward) using quantifier laws:
$$\forall x\in\mathbb{R}\ \exists y\in\mathbb{R}\text{ such that } y>x+174.$$
(a) Give the symbolic negation step-by-step.
(b) Translate both the original statement and its negation into plain English... | [
{
"method_name": "English Semantics Cross-Check",
"approach": "Translate to English, negate carefully, then translate back to symbols.",
"steps": [
"Step 1: Original: 'For every real $x$, there exists a real $y$ that is greater than $x+174$.'",
"Step 2: Negation: 'There exists a real $x$ suc... | {
"consistency_check": "Both approaches agree after simplification. Final answer: $\\boxed{\\exists x\\in\\mathbb{R}$.\nThe formal quantifier-negation rules and the English semantics yield the same statement: the existence of an $x$ that upper-bounds all reals after shifting by a constant.",
"robustness_analysis": ... | [
{
"error_description": "Negated only the inequality but kept quantifiers unchanged.",
"why_plausible": "The inequality is the most visible part of the statement, so attention focuses there.",
"why_wrong": "Quantifiers determine scope; the correct negation flips $\\forall\\leftrightarrow\\exists$ and cha... | Key idea: Negating quantified statements requires flipping each quantifier and negating the predicate while keeping variable order. Always sanity-check by translating to plain English. |
math-001942 | Foundations: Translating Statements | 1 | Carefully track domains: Negate the statement and simplify (push negation inward) using quantifier laws:
$$\forall x\in\mathbb{R}\ \exists y\in\mathbb{R}\text{ such that } y>x+393.$$
(a) Give the symbolic negation step-by-step.
(b) Translate both the original statement and its negation into plain English to confirm th... | [
{
"method_name": "Quantifier Laws",
"approach": "Use $\\neg\\forall=\\exists\\neg$ and $\\neg\\exists=\\forall\\neg$, then negate the inequality.",
"steps": [
"Step 1: Start: $\\neg(\\forall x\\ \\exists y\\ P(x,y))$.",
"Step 2: Apply $\\neg\\forall x\\ Q(x)\\equiv \\exists x\\ \\neg Q(x)$ t... | {
"consistency_check": "The two methods are consistent and must coincide. Final answer: $\\boxed{\\exists x\\in\\mathbb{R}$.\nThe formal quantifier-negation rules and the English semantics yield the same statement: the existence of an $x$ that upper-bounds all reals after shifting by a constant.",
"robustness_analy... | [
{
"error_description": "Negated only the inequality but kept quantifiers unchanged.",
"why_plausible": "The inequality is the most visible part of the statement, so attention focuses there.",
"why_wrong": "Quantifiers determine scope; the correct negation flips $\\forall\\leftrightarrow\\exists$ and cha... | Takeaway: Negating quantified statements requires flipping each quantifier and negating the predicate while keeping variable order. Always sanity-check by translating to plain English. |
math-001943 | Logic: Formal vs English Paraphrase | 1 | Determine the requested value: Negate the statement and simplify (push negation inward) using quantifier laws:
$$\forall x\in\mathbb{R}\ \exists y\in\mathbb{R}\text{ such that } y>x+295.$$
(a) Give the symbolic negation step-by-step.
(b) Translate both the original statement and its negation into plain English to conf... | [
{
"method_name": "English Semantics Cross-Check",
"approach": "Translate to English, negate carefully, then translate back to symbols.",
"steps": [
"Step 1: Original: 'For every real $x$, there exists a real $y$ that is greater than $x+295$.'",
"Step 2: Negation: 'There exists a real $x$ suc... | {
"consistency_check": "Both approaches agree after simplification. Final answer: $\\boxed{\\exists x\\in\\mathbb{R}$.\nThe formal quantifier-negation rules and the English semantics yield the same statement: the existence of an $x$ that upper-bounds all reals after shifting by a constant.",
"robustness_analysis": ... | [
{
"error_description": "Negated only the inequality but kept quantifiers unchanged.",
"why_plausible": "The inequality is the most visible part of the statement, so attention focuses there.",
"why_wrong": "Quantifiers determine scope; the correct negation flips $\\forall\\leftrightarrow\\exists$ and cha... | Takeaway: Negating quantified statements requires flipping each quantifier and negating the predicate while keeping variable order. Always sanity-check by translating to plain English. (Here the result is $\boxed{\exists x\in\mathbb{R}$.) |
math-001944 | Logic: Quantifiers — Negation Rules | 1 | Carefully track domains: Negate the statement and simplify (push negation inward) using quantifier laws:
$$\forall x\in\mathbb{R}\ \exists y\in\mathbb{R}\text{ such that } y>x+228.$$
(a) Give the symbolic negation step-by-step.
(b) Translate both the original statement and its negation into plain English to confirm th... | [
{
"method_name": "Quantifier Laws",
"approach": "Use $\\neg\\forall=\\exists\\neg$ and $\\neg\\exists=\\forall\\neg$, then negate the inequality.",
"steps": [
"Step 1: Start: $\\neg(\\forall x\\ \\exists y\\ P(x,y))$.",
"Step 2: Apply $\\neg\\forall x\\ Q(x)\\equiv \\exists x\\ \\neg Q(x)$ t... | {
"consistency_check": "Cross-check: both derivations land on the same invariant quantity. Final answer: $\\boxed{\\exists x\\in\\mathbb{R}$.\nThe formal quantifier-negation rules and the English semantics yield the same statement: the existence of an $x$ that upper-bounds all reals after shifting by a constant.",
... | [
{
"error_description": "Negated only the inequality but kept quantifiers unchanged.",
"why_plausible": "The inequality is the most visible part of the statement, so attention focuses there.",
"why_wrong": "Quantifiers determine scope; the correct negation flips $\\forall\\leftrightarrow\\exists$ and cha... | Remember: Negating quantified statements requires flipping each quantifier and negating the predicate while keeping variable order. Always sanity-check by translating to plain English. (Here the result is $\boxed{\exists x\in\mathbb{R}$.) |
math-001945 | Logic: Quantifiers — Negation Rules | 1 | Explain each transformation: Negate the statement and simplify (push negation inward) using quantifier laws:
$$\forall x\in\mathbb{R}\ \exists y\in\mathbb{R}\text{ such that } y>x+38.$$
(a) Give the symbolic negation step-by-step.
(b) Translate both the original statement and its negation into plain English to confirm... | [
{
"method_name": "Quantifier Laws",
"approach": "Use $\\neg\\forall=\\exists\\neg$ and $\\neg\\exists=\\forall\\neg$, then negate the inequality.",
"steps": [
"Step 1: Start: $\\neg(\\forall x\\ \\exists y\\ P(x,y))$.",
"Step 2: Apply $\\neg\\forall x\\ Q(x)\\equiv \\exists x\\ \\neg Q(x)$ t... | {
"consistency_check": "The two methods are consistent and must coincide. Final answer: $\\boxed{\\exists x\\in\\mathbb{R}$.\nThe formal quantifier-negation rules and the English semantics yield the same statement: the existence of an $x$ that upper-bounds all reals after shifting by a constant.",
"robustness_analy... | [
{
"error_description": "Negated only the inequality but kept quantifiers unchanged.",
"why_plausible": "The inequality is the most visible part of the statement, so attention focuses there.",
"why_wrong": "Quantifiers determine scope; the correct negation flips $\\forall\\leftrightarrow\\exists$ and cha... | Takeaway: Negating quantified statements requires flipping each quantifier and negating the predicate while keeping variable order. Always sanity-check by translating to plain English. |
math-001946 | Logic: Scope — Order of Quantifiers | 1 | Prompt: Negate the statement and simplify (push negation inward) using quantifier laws:
$$\forall x\in\mathbb{R}\ \exists y\in\mathbb{R}\text{ such that } y>x+173.$$
(a) Give the symbolic negation step-by-step.
(b) Translate both the original statement and its negation into plain English to confirm the meaning.
Inclu... | [
{
"method_name": "English Semantics Cross-Check",
"approach": "Translate to English, negate carefully, then translate back to symbols.",
"steps": [
"Step 1: Original: 'For every real $x$, there exists a real $y$ that is greater than $x+173$.'",
"Step 2: Negation: 'There exists a real $x$ suc... | {
"consistency_check": "Both approaches agree after simplification. Final answer: $\\boxed{\\exists x\\in\\mathbb{R}$.\nThe formal quantifier-negation rules and the English semantics yield the same statement: the existence of an $x$ that upper-bounds all reals after shifting by a constant.",
"robustness_analysis": ... | [
{
"error_description": "Negated only the inequality but kept quantifiers unchanged.",
"why_plausible": "The inequality is the most visible part of the statement, so attention focuses there.",
"why_wrong": "Quantifiers determine scope; the correct negation flips $\\forall\\leftrightarrow\\exists$ and cha... | Key idea: Negating quantified statements requires flipping each quantifier and negating the predicate while keeping variable order. Always sanity-check by translating to plain English. |
math-001947 | Foundations: Translating Statements | 1 | Prompt: Negate the statement and simplify (push negation inward) using quantifier laws:
$$\forall x\in\mathbb{R}\ \exists y\in\mathbb{R}\text{ such that } y>x+36.$$
(a) Give the symbolic negation step-by-step.
(b) Translate both the original statement and its negation into plain English to confirm the meaning.
Includ... | [
{
"method_name": "Quantifier Laws",
"approach": "Use $\\neg\\forall=\\exists\\neg$ and $\\neg\\exists=\\forall\\neg$, then negate the inequality.",
"steps": [
"Step 1: Start: $\\neg(\\forall x\\ \\exists y\\ P(x,y))$.",
"Step 2: Apply $\\neg\\forall x\\ Q(x)\\equiv \\exists x\\ \\neg Q(x)$ t... | {
"consistency_check": "Consistency verification shows both paths yield the identical boxed result. Final answer: $\\boxed{\\exists x\\in\\mathbb{R}$.\nThe formal quantifier-negation rules and the English semantics yield the same statement: the existence of an $x$ that upper-bounds all reals after shifting by a const... | [
{
"error_description": "Negated only the inequality but kept quantifiers unchanged.",
"why_plausible": "The inequality is the most visible part of the statement, so attention focuses there.",
"why_wrong": "Quantifiers determine scope; the correct negation flips $\\forall\\leftrightarrow\\exists$ and cha... | Key idea: Negating quantified statements requires flipping each quantifier and negating the predicate while keeping variable order. Always sanity-check by translating to plain English. (Here the result is $\boxed{\exists x\in\mathbb{R}$.) |
math-001948 | Logic: Formal vs English Paraphrase | 1 | Provide a rigorous solution: Negate the statement and simplify (push negation inward) using quantifier laws:
$$\forall x\in\mathbb{R}\ \exists y\in\mathbb{R}\text{ such that } y>x+230.$$
(a) Give the symbolic negation step-by-step.
(b) Translate both the original statement and its negation into plain English to confir... | [
{
"method_name": "English Semantics Cross-Check",
"approach": "Translate to English, negate carefully, then translate back to symbols.",
"steps": [
"Step 1: Original: 'For every real $x$, there exists a real $y$ that is greater than $x+230$.'",
"Step 2: Negation: 'There exists a real $x$ suc... | {
"consistency_check": "Both approaches agree after simplification. Final answer: $\\boxed{\\exists x\\in\\mathbb{R}$.\nThe formal quantifier-negation rules and the English semantics yield the same statement: the existence of an $x$ that upper-bounds all reals after shifting by a constant.",
"robustness_analysis": ... | [
{
"error_description": "Negated only the inequality but kept quantifiers unchanged.",
"why_plausible": "The inequality is the most visible part of the statement, so attention focuses there.",
"why_wrong": "Quantifiers determine scope; the correct negation flips $\\forall\\leftrightarrow\\exists$ and cha... | Takeaway: Negating quantified statements requires flipping each quantifier and negating the predicate while keeping variable order. Always sanity-check by translating to plain English. |
math-001949 | Foundations: Translating Statements | 1 | Complete the analysis: Negate the statement and simplify (push negation inward) using quantifier laws:
$$\forall x\in\mathbb{R}\ \exists y\in\mathbb{R}\text{ such that } y>x+214.$$
(a) Give the symbolic negation step-by-step.
(b) Translate both the original statement and its negation into plain English to confirm the ... | [
{
"method_name": "English Semantics Cross-Check",
"approach": "Translate to English, negate carefully, then translate back to symbols.",
"steps": [
"Step 1: Original: 'For every real $x$, there exists a real $y$ that is greater than $x+214$.'",
"Step 2: Negation: 'There exists a real $x$ suc... | {
"consistency_check": "The two methods are consistent and must coincide. Final answer: $\\boxed{\\exists x\\in\\mathbb{R}$.\nThe formal quantifier-negation rules and the English semantics yield the same statement: the existence of an $x$ that upper-bounds all reals after shifting by a constant.",
"robustness_analy... | [
{
"error_description": "Negated only the inequality but kept quantifiers unchanged.",
"why_plausible": "The inequality is the most visible part of the statement, so attention focuses there.",
"why_wrong": "Quantifiers determine scope; the correct negation flips $\\forall\\leftrightarrow\\exists$ and cha... | Takeaway: Negating quantified statements requires flipping each quantifier and negating the predicate while keeping variable order. Always sanity-check by translating to plain English. |
math-001950 | Logic: Formal vs English Paraphrase | 1 | Derive the result step-by-step: Negate the statement and simplify (push negation inward) using quantifier laws:
$$\forall x\in\mathbb{R}\ \exists y\in\mathbb{R}\text{ such that } y>x+128.$$
(a) Give the symbolic negation step-by-step.
(b) Translate both the original statement and its negation into plain English to con... | [
{
"method_name": "English Semantics Cross-Check",
"approach": "Translate to English, negate carefully, then translate back to symbols.",
"steps": [
"Step 1: Original: 'For every real $x$, there exists a real $y$ that is greater than $x+128$.'",
"Step 2: Negation: 'There exists a real $x$ suc... | {
"consistency_check": "Cross-check: both derivations land on the same invariant quantity. Final answer: $\\boxed{\\exists x\\in\\mathbb{R}$.\nThe formal quantifier-negation rules and the English semantics yield the same statement: the existence of an $x$ that upper-bounds all reals after shifting by a constant.",
... | [
{
"error_description": "Negated only the inequality but kept quantifiers unchanged.",
"why_plausible": "The inequality is the most visible part of the statement, so attention focuses there.",
"why_wrong": "Quantifiers determine scope; the correct negation flips $\\forall\\leftrightarrow\\exists$ and cha... | Key idea: Negating quantified statements requires flipping each quantifier and negating the predicate while keeping variable order. Always sanity-check by translating to plain English. |
math-001951 | Logic: Formal vs English Paraphrase | 1 | Compute the requested quantity: Negate the statement and simplify (push negation inward) using quantifier laws:
$$\forall x\in\mathbb{R}\ \exists y\in\mathbb{R}\text{ such that } y>x+148.$$
(a) Give the symbolic negation step-by-step.
(b) Translate both the original statement and its negation into plain English to con... | [
{
"method_name": "Quantifier Laws",
"approach": "Use $\\neg\\forall=\\exists\\neg$ and $\\neg\\exists=\\forall\\neg$, then negate the inequality.",
"steps": [
"Step 1: Start: $\\neg(\\forall x\\ \\exists y\\ P(x,y))$.",
"Step 2: Apply $\\neg\\forall x\\ Q(x)\\equiv \\exists x\\ \\neg Q(x)$ t... | {
"consistency_check": "Both approaches agree after simplification. Final answer: $\\boxed{\\exists x\\in\\mathbb{R}$.\nThe formal quantifier-negation rules and the English semantics yield the same statement: the existence of an $x$ that upper-bounds all reals after shifting by a constant.",
"robustness_analysis": ... | [
{
"error_description": "Negated only the inequality but kept quantifiers unchanged.",
"why_plausible": "The inequality is the most visible part of the statement, so attention focuses there.",
"why_wrong": "Quantifiers determine scope; the correct negation flips $\\forall\\leftrightarrow\\exists$ and cha... | Remember: Negating quantified statements requires flipping each quantifier and negating the predicate while keeping variable order. Always sanity-check by translating to plain English. (Here the result is $\boxed{\exists x\in\mathbb{R}$.) |
math-001952 | Logic: Scope — Order of Quantifiers | 1 | Provide a rigorous solution: Negate the statement and simplify (push negation inward) using quantifier laws:
$$\forall x\in\mathbb{R}\ \exists y\in\mathbb{R}\text{ such that } y>x+392.$$
(a) Give the symbolic negation step-by-step.
(b) Translate both the original statement and its negation into plain English to confir... | [
{
"method_name": "English Semantics Cross-Check",
"approach": "Translate to English, negate carefully, then translate back to symbols.",
"steps": [
"Step 1: Original: 'For every real $x$, there exists a real $y$ that is greater than $x+392$.'",
"Step 2: Negation: 'There exists a real $x$ suc... | {
"consistency_check": "Both approaches agree after simplification. Final answer: $\\boxed{\\exists x\\in\\mathbb{R}$.\nThe formal quantifier-negation rules and the English semantics yield the same statement: the existence of an $x$ that upper-bounds all reals after shifting by a constant.",
"robustness_analysis": ... | [
{
"error_description": "Negated only the inequality but kept quantifiers unchanged.",
"why_plausible": "The inequality is the most visible part of the statement, so attention focuses there.",
"why_wrong": "Quantifiers determine scope; the correct negation flips $\\forall\\leftrightarrow\\exists$ and cha... | Takeaway: Negating quantified statements requires flipping each quantifier and negating the predicate while keeping variable order. Always sanity-check by translating to plain English. (Here the result is $\boxed{\exists x\in\mathbb{R}$.) |
math-001953 | Foundations: Translating Statements | 1 | Explain why your operations are valid: Negate the statement and simplify (push negation inward) using quantifier laws:
$$\forall x\in\mathbb{R}\ \exists y\in\mathbb{R}\text{ such that } y>x+330.$$
(a) Give the symbolic negation step-by-step.
(b) Translate both the original statement and its negation into plain English... | [
{
"method_name": "English Semantics Cross-Check",
"approach": "Translate to English, negate carefully, then translate back to symbols.",
"steps": [
"Step 1: Original: 'For every real $x$, there exists a real $y$ that is greater than $x+330$.'",
"Step 2: Negation: 'There exists a real $x$ suc... | {
"consistency_check": "Consistency verification shows both paths yield the identical boxed result. Final answer: $\\boxed{\\exists x\\in\\mathbb{R}$.\nThe formal quantifier-negation rules and the English semantics yield the same statement: the existence of an $x$ that upper-bounds all reals after shifting by a const... | [
{
"error_description": "Negated only the inequality but kept quantifiers unchanged.",
"why_plausible": "The inequality is the most visible part of the statement, so attention focuses there.",
"why_wrong": "Quantifiers determine scope; the correct negation flips $\\forall\\leftrightarrow\\exists$ and cha... | Core principle: Negating quantified statements requires flipping each quantifier and negating the predicate while keeping variable order. Always sanity-check by translating to plain English. (Here the result is $\boxed{\exists x\in\mathbb{R}$.) |
math-001954 | Logic: Scope — Order of Quantifiers | 1 | Checkpoint: Negate the statement and simplify (push negation inward) using quantifier laws:
$$\forall x\in\mathbb{R}\ \exists y\in\mathbb{R}\text{ such that } y>x+170.$$
(a) Give the symbolic negation step-by-step.
(b) Translate both the original statement and its negation into plain English to confirm the meaning.
I... | [
{
"method_name": "Quantifier Laws",
"approach": "Use $\\neg\\forall=\\exists\\neg$ and $\\neg\\exists=\\forall\\neg$, then negate the inequality.",
"steps": [
"Step 1: Start: $\\neg(\\forall x\\ \\exists y\\ P(x,y))$.",
"Step 2: Apply $\\neg\\forall x\\ Q(x)\\equiv \\exists x\\ \\neg Q(x)$ t... | {
"consistency_check": "Both approaches agree after simplification. Final answer: $\\boxed{\\exists x\\in\\mathbb{R}$.\nThe formal quantifier-negation rules and the English semantics yield the same statement: the existence of an $x$ that upper-bounds all reals after shifting by a constant.",
"robustness_analysis": ... | [
{
"error_description": "Negated only the inequality but kept quantifiers unchanged.",
"why_plausible": "The inequality is the most visible part of the statement, so attention focuses there.",
"why_wrong": "Quantifiers determine scope; the correct negation flips $\\forall\\leftrightarrow\\exists$ and cha... | Takeaway: Negating quantified statements requires flipping each quantifier and negating the predicate while keeping variable order. Always sanity-check by translating to plain English. (Here the result is $\boxed{\exists x\in\mathbb{R}$.) |
math-001955 | Logic: Formal vs English Paraphrase | 1 | Challenge: Negate the statement and simplify (push negation inward) using quantifier laws:
$$\forall x\in\mathbb{R}\ \exists y\in\mathbb{R}\text{ such that } y>x+135.$$
(a) Give the symbolic negation step-by-step.
(b) Translate both the original statement and its negation into plain English to confirm the meaning.
In... | [
{
"method_name": "English Semantics Cross-Check",
"approach": "Translate to English, negate carefully, then translate back to symbols.",
"steps": [
"Step 1: Original: 'For every real $x$, there exists a real $y$ that is greater than $x+135$.'",
"Step 2: Negation: 'There exists a real $x$ suc... | {
"consistency_check": "Both approaches agree after simplification. Final answer: $\\boxed{\\exists x\\in\\mathbb{R}$.\nThe formal quantifier-negation rules and the English semantics yield the same statement: the existence of an $x$ that upper-bounds all reals after shifting by a constant.",
"robustness_analysis": ... | [
{
"error_description": "Negated only the inequality but kept quantifiers unchanged.",
"why_plausible": "The inequality is the most visible part of the statement, so attention focuses there.",
"why_wrong": "Quantifiers determine scope; the correct negation flips $\\forall\\leftrightarrow\\exists$ and cha... | Remember: Negating quantified statements requires flipping each quantifier and negating the predicate while keeping variable order. Always sanity-check by translating to plain English. |
math-001956 | Logic: Scope — Order of Quantifiers | 1 | Solve and justify each step: Negate the statement and simplify (push negation inward) using quantifier laws:
$$\forall x\in\mathbb{R}\ \exists y\in\mathbb{R}\text{ such that } y>x+193.$$
(a) Give the symbolic negation step-by-step.
(b) Translate both the original statement and its negation into plain English to confir... | [
{
"method_name": "English Semantics Cross-Check",
"approach": "Translate to English, negate carefully, then translate back to symbols.",
"steps": [
"Step 1: Original: 'For every real $x$, there exists a real $y$ that is greater than $x+193$.'",
"Step 2: Negation: 'There exists a real $x$ suc... | {
"consistency_check": "Consistency verification shows both paths yield the identical boxed result. Final answer: $\\boxed{\\exists x\\in\\mathbb{R}$.\nThe formal quantifier-negation rules and the English semantics yield the same statement: the existence of an $x$ that upper-bounds all reals after shifting by a const... | [
{
"error_description": "Negated only the inequality but kept quantifiers unchanged.",
"why_plausible": "The inequality is the most visible part of the statement, so attention focuses there.",
"why_wrong": "Quantifiers determine scope; the correct negation flips $\\forall\\leftrightarrow\\exists$ and cha... | Key idea: Negating quantified statements requires flipping each quantifier and negating the predicate while keeping variable order. Always sanity-check by translating to plain English. (Here the result is $\boxed{\exists x\in\mathbb{R}$.) |
math-001957 | Logic: Scope — Order of Quantifiers | 1 | Give an answer and a quick verification: Negate the statement and simplify (push negation inward) using quantifier laws:
$$\forall x\in\mathbb{R}\ \exists y\in\mathbb{R}\text{ such that } y>x+303.$$
(a) Give the symbolic negation step-by-step.
(b) Translate both the original statement and its negation into plain Engli... | [
{
"method_name": "English Semantics Cross-Check",
"approach": "Translate to English, negate carefully, then translate back to symbols.",
"steps": [
"Step 1: Original: 'For every real $x$, there exists a real $y$ that is greater than $x+303$.'",
"Step 2: Negation: 'There exists a real $x$ suc... | {
"consistency_check": "The two methods are consistent and must coincide. Final answer: $\\boxed{\\exists x\\in\\mathbb{R}$.\nThe formal quantifier-negation rules and the English semantics yield the same statement: the existence of an $x$ that upper-bounds all reals after shifting by a constant.",
"robustness_analy... | [
{
"error_description": "Negated only the inequality but kept quantifiers unchanged.",
"why_plausible": "The inequality is the most visible part of the statement, so attention focuses there.",
"why_wrong": "Quantifiers determine scope; the correct negation flips $\\forall\\leftrightarrow\\exists$ and cha... | Remember: Negating quantified statements requires flipping each quantifier and negating the predicate while keeping variable order. Always sanity-check by translating to plain English. |
math-001958 | Foundations: Translating Statements | 1 | Track quantifiers carefully: Negate the statement and simplify (push negation inward) using quantifier laws:
$$\forall x\in\mathbb{R}\ \exists y\in\mathbb{R}\text{ such that } y>x+310.$$
(a) Give the symbolic negation step-by-step.
(b) Translate both the original statement and its negation into plain English to confir... | [
{
"method_name": "Quantifier Laws",
"approach": "Use $\\neg\\forall=\\exists\\neg$ and $\\neg\\exists=\\forall\\neg$, then negate the inequality.",
"steps": [
"Step 1: Start: $\\neg(\\forall x\\ \\exists y\\ P(x,y))$.",
"Step 2: Apply $\\neg\\forall x\\ Q(x)\\equiv \\exists x\\ \\neg Q(x)$ t... | {
"consistency_check": "Consistency verification shows both paths yield the identical boxed result. Final answer: $\\boxed{\\exists x\\in\\mathbb{R}$.\nThe formal quantifier-negation rules and the English semantics yield the same statement: the existence of an $x$ that upper-bounds all reals after shifting by a const... | [
{
"error_description": "Negated only the inequality but kept quantifiers unchanged.",
"why_plausible": "The inequality is the most visible part of the statement, so attention focuses there.",
"why_wrong": "Quantifiers determine scope; the correct negation flips $\\forall\\leftrightarrow\\exists$ and cha... | Key idea: Negating quantified statements requires flipping each quantifier and negating the predicate while keeping variable order. Always sanity-check by translating to plain English. (Here the result is $\boxed{\exists x\in\mathbb{R}$.) |
math-001959 | Logic: Scope — Order of Quantifiers | 1 | Compute the requested quantity: Negate the statement and simplify (push negation inward) using quantifier laws:
$$\forall x\in\mathbb{R}\ \exists y\in\mathbb{R}\text{ such that } y>x+250.$$
(a) Give the symbolic negation step-by-step.
(b) Translate both the original statement and its negation into plain English to con... | [
{
"method_name": "Quantifier Laws",
"approach": "Use $\\neg\\forall=\\exists\\neg$ and $\\neg\\exists=\\forall\\neg$, then negate the inequality.",
"steps": [
"Step 1: Start: $\\neg(\\forall x\\ \\exists y\\ P(x,y))$.",
"Step 2: Apply $\\neg\\forall x\\ Q(x)\\equiv \\exists x\\ \\neg Q(x)$ t... | {
"consistency_check": "The two methods are consistent and must coincide. Final answer: $\\boxed{\\exists x\\in\\mathbb{R}$.\nThe formal quantifier-negation rules and the English semantics yield the same statement: the existence of an $x$ that upper-bounds all reals after shifting by a constant.",
"robustness_analy... | [
{
"error_description": "Negated only the inequality but kept quantifiers unchanged.",
"why_plausible": "The inequality is the most visible part of the statement, so attention focuses there.",
"why_wrong": "Quantifiers determine scope; the correct negation flips $\\forall\\leftrightarrow\\exists$ and cha... | Key idea: Negating quantified statements requires flipping each quantifier and negating the predicate while keeping variable order. Always sanity-check by translating to plain English. (Here the result is $\boxed{\exists x\in\mathbb{R}$.) |
math-001960 | Logic: Scope — Order of Quantifiers | 1 | Solve and include a self-check: Negate the statement and simplify (push negation inward) using quantifier laws:
$$\forall x\in\mathbb{R}\ \exists y\in\mathbb{R}\text{ such that } y>x+144.$$
(a) Give the symbolic negation step-by-step.
(b) Translate both the original statement and its negation into plain English to con... | [
{
"method_name": "English Semantics Cross-Check",
"approach": "Translate to English, negate carefully, then translate back to symbols.",
"steps": [
"Step 1: Original: 'For every real $x$, there exists a real $y$ that is greater than $x+144$.'",
"Step 2: Negation: 'There exists a real $x$ suc... | {
"consistency_check": "Consistency verification shows both paths yield the identical boxed result. Final answer: $\\boxed{\\exists x\\in\\mathbb{R}$.\nThe formal quantifier-negation rules and the English semantics yield the same statement: the existence of an $x$ that upper-bounds all reals after shifting by a const... | [
{
"error_description": "Negated only the inequality but kept quantifiers unchanged.",
"why_plausible": "The inequality is the most visible part of the statement, so attention focuses there.",
"why_wrong": "Quantifiers determine scope; the correct negation flips $\\forall\\leftrightarrow\\exists$ and cha... | Takeaway: Negating quantified statements requires flipping each quantifier and negating the predicate while keeping variable order. Always sanity-check by translating to plain English. |
math-001961 | Logic: Formal vs English Paraphrase | 1 | Work this out carefully: Negate the statement and simplify (push negation inward) using quantifier laws:
$$\forall x\in\mathbb{R}\ \exists y\in\mathbb{R}\text{ such that } y>x+281.$$
(a) Give the symbolic negation step-by-step.
(b) Translate both the original statement and its negation into plain English to confirm th... | [
{
"method_name": "Quantifier Laws",
"approach": "Use $\\neg\\forall=\\exists\\neg$ and $\\neg\\exists=\\forall\\neg$, then negate the inequality.",
"steps": [
"Step 1: Start: $\\neg(\\forall x\\ \\exists y\\ P(x,y))$.",
"Step 2: Apply $\\neg\\forall x\\ Q(x)\\equiv \\exists x\\ \\neg Q(x)$ t... | {
"consistency_check": "Consistency verification shows both paths yield the identical boxed result. Final answer: $\\boxed{\\exists x\\in\\mathbb{R}$.\nThe formal quantifier-negation rules and the English semantics yield the same statement: the existence of an $x$ that upper-bounds all reals after shifting by a const... | [
{
"error_description": "Negated only the inequality but kept quantifiers unchanged.",
"why_plausible": "The inequality is the most visible part of the statement, so attention focuses there.",
"why_wrong": "Quantifiers determine scope; the correct negation flips $\\forall\\leftrightarrow\\exists$ and cha... | Remember: Negating quantified statements requires flipping each quantifier and negating the predicate while keeping variable order. Always sanity-check by translating to plain English. (Here the result is $\boxed{\exists x\in\mathbb{R}$.) |
math-001962 | Logic: Formal vs English Paraphrase | 1 | Warm-up: Negate the statement and simplify (push negation inward) using quantifier laws:
$$\forall x\in\mathbb{R}\ \exists y\in\mathbb{R}\text{ such that } y>x+225.$$
(a) Give the symbolic negation step-by-step.
(b) Translate both the original statement and its negation into plain English to confirm the meaning.
Incl... | [
{
"method_name": "Quantifier Laws",
"approach": "Use $\\neg\\forall=\\exists\\neg$ and $\\neg\\exists=\\forall\\neg$, then negate the inequality.",
"steps": [
"Step 1: Start: $\\neg(\\forall x\\ \\exists y\\ P(x,y))$.",
"Step 2: Apply $\\neg\\forall x\\ Q(x)\\equiv \\exists x\\ \\neg Q(x)$ t... | {
"consistency_check": "Cross-check: both derivations land on the same invariant quantity. Final answer: $\\boxed{\\exists x\\in\\mathbb{R}$.\nThe formal quantifier-negation rules and the English semantics yield the same statement: the existence of an $x$ that upper-bounds all reals after shifting by a constant.",
... | [
{
"error_description": "Negated only the inequality but kept quantifiers unchanged.",
"why_plausible": "The inequality is the most visible part of the statement, so attention focuses there.",
"why_wrong": "Quantifiers determine scope; the correct negation flips $\\forall\\leftrightarrow\\exists$ and cha... | Remember: Negating quantified statements requires flipping each quantifier and negating the predicate while keeping variable order. Always sanity-check by translating to plain English. (Here the result is $\boxed{\exists x\in\mathbb{R}$.) |
math-001963 | Logic: Scope — Order of Quantifiers | 1 | Explain why your operations are valid: Negate the statement and simplify (push negation inward) using quantifier laws:
$$\forall x\in\mathbb{R}\ \exists y\in\mathbb{R}\text{ such that } y>x+353.$$
(a) Give the symbolic negation step-by-step.
(b) Translate both the original statement and its negation into plain English... | [
{
"method_name": "English Semantics Cross-Check",
"approach": "Translate to English, negate carefully, then translate back to symbols.",
"steps": [
"Step 1: Original: 'For every real $x$, there exists a real $y$ that is greater than $x+353$.'",
"Step 2: Negation: 'There exists a real $x$ suc... | {
"consistency_check": "Consistency verification shows both paths yield the identical boxed result. Final answer: $\\boxed{\\exists x\\in\\mathbb{R}$.\nThe formal quantifier-negation rules and the English semantics yield the same statement: the existence of an $x$ that upper-bounds all reals after shifting by a const... | [
{
"error_description": "Negated only the inequality but kept quantifiers unchanged.",
"why_plausible": "The inequality is the most visible part of the statement, so attention focuses there.",
"why_wrong": "Quantifiers determine scope; the correct negation flips $\\forall\\leftrightarrow\\exists$ and cha... | Remember: Negating quantified statements requires flipping each quantifier and negating the predicate while keeping variable order. Always sanity-check by translating to plain English. (Here the result is $\boxed{\exists x\in\mathbb{R}$.) |
math-001964 | Logic: Quantifiers — Negation Rules | 1 | Warm-up: Negate the statement and simplify (push negation inward) using quantifier laws:
$$\forall x\in\mathbb{R}\ \exists y\in\mathbb{R}\text{ such that } y>x+354.$$
(a) Give the symbolic negation step-by-step.
(b) Translate both the original statement and its negation into plain English to confirm the meaning.
Incl... | [
{
"method_name": "English Semantics Cross-Check",
"approach": "Translate to English, negate carefully, then translate back to symbols.",
"steps": [
"Step 1: Original: 'For every real $x$, there exists a real $y$ that is greater than $x+354$.'",
"Step 2: Negation: 'There exists a real $x$ suc... | {
"consistency_check": "The two methods are consistent and must coincide. Final answer: $\\boxed{\\exists x\\in\\mathbb{R}$.\nThe formal quantifier-negation rules and the English semantics yield the same statement: the existence of an $x$ that upper-bounds all reals after shifting by a constant.",
"robustness_analy... | [
{
"error_description": "Negated only the inequality but kept quantifiers unchanged.",
"why_plausible": "The inequality is the most visible part of the statement, so attention focuses there.",
"why_wrong": "Quantifiers determine scope; the correct negation flips $\\forall\\leftrightarrow\\exists$ and cha... | Core principle: Negating quantified statements requires flipping each quantifier and negating the predicate while keeping variable order. Always sanity-check by translating to plain English. |
math-001965 | Logic: Scope — Order of Quantifiers | 1 | Solve and include a self-check: Negate the statement and simplify (push negation inward) using quantifier laws:
$$\forall x\in\mathbb{R}\ \exists y\in\mathbb{R}\text{ such that } y>x+141.$$
(a) Give the symbolic negation step-by-step.
(b) Translate both the original statement and its negation into plain English to con... | [
{
"method_name": "English Semantics Cross-Check",
"approach": "Translate to English, negate carefully, then translate back to symbols.",
"steps": [
"Step 1: Original: 'For every real $x$, there exists a real $y$ that is greater than $x+141$.'",
"Step 2: Negation: 'There exists a real $x$ suc... | {
"consistency_check": "Consistency verification shows both paths yield the identical boxed result. Final answer: $\\boxed{\\exists x\\in\\mathbb{R}$.\nThe formal quantifier-negation rules and the English semantics yield the same statement: the existence of an $x$ that upper-bounds all reals after shifting by a const... | [
{
"error_description": "Negated only the inequality but kept quantifiers unchanged.",
"why_plausible": "The inequality is the most visible part of the statement, so attention focuses there.",
"why_wrong": "Quantifiers determine scope; the correct negation flips $\\forall\\leftrightarrow\\exists$ and cha... | Core principle: Negating quantified statements requires flipping each quantifier and negating the predicate while keeping variable order. Always sanity-check by translating to plain English. |
math-001966 | Logic: Formal vs English Paraphrase | 1 | Work carefully and justify each inference: Negate the statement and simplify (push negation inward) using quantifier laws:
$$\forall x\in\mathbb{R}\ \exists y\in\mathbb{R}\text{ such that } y>x+344.$$
(a) Give the symbolic negation step-by-step.
(b) Translate both the original statement and its negation into plain Eng... | [
{
"method_name": "English Semantics Cross-Check",
"approach": "Translate to English, negate carefully, then translate back to symbols.",
"steps": [
"Step 1: Original: 'For every real $x$, there exists a real $y$ that is greater than $x+344$.'",
"Step 2: Negation: 'There exists a real $x$ suc... | {
"consistency_check": "Both approaches agree after simplification. Final answer: $\\boxed{\\exists x\\in\\mathbb{R}$.\nThe formal quantifier-negation rules and the English semantics yield the same statement: the existence of an $x$ that upper-bounds all reals after shifting by a constant.",
"robustness_analysis": ... | [
{
"error_description": "Negated only the inequality but kept quantifiers unchanged.",
"why_plausible": "The inequality is the most visible part of the statement, so attention focuses there.",
"why_wrong": "Quantifiers determine scope; the correct negation flips $\\forall\\leftrightarrow\\exists$ and cha... | Remember: Negating quantified statements requires flipping each quantifier and negating the predicate while keeping variable order. Always sanity-check by translating to plain English. (Here the result is $\boxed{\exists x\in\mathbb{R}$.) |
math-001967 | Logic: Scope — Order of Quantifiers | 1 | Track quantifiers carefully: Negate the statement and simplify (push negation inward) using quantifier laws:
$$\forall x\in\mathbb{R}\ \exists y\in\mathbb{R}\text{ such that } y>x+26.$$
(a) Give the symbolic negation step-by-step.
(b) Translate both the original statement and its negation into plain English to confirm... | [
{
"method_name": "English Semantics Cross-Check",
"approach": "Translate to English, negate carefully, then translate back to symbols.",
"steps": [
"Step 1: Original: 'For every real $x$, there exists a real $y$ that is greater than $x+26$.'",
"Step 2: Negation: 'There exists a real $x$ such... | {
"consistency_check": "Both approaches agree after simplification. Final answer: $\\boxed{\\exists x\\in\\mathbb{R}$.\nThe formal quantifier-negation rules and the English semantics yield the same statement: the existence of an $x$ that upper-bounds all reals after shifting by a constant.",
"robustness_analysis": ... | [
{
"error_description": "Negated only the inequality but kept quantifiers unchanged.",
"why_plausible": "The inequality is the most visible part of the statement, so attention focuses there.",
"why_wrong": "Quantifiers determine scope; the correct negation flips $\\forall\\leftrightarrow\\exists$ and cha... | Key idea: Negating quantified statements requires flipping each quantifier and negating the predicate while keeping variable order. Always sanity-check by translating to plain English. |
math-001968 | Foundations: Translating Statements | 1 | Solve with verification: Negate the statement and simplify (push negation inward) using quantifier laws:
$$\forall x\in\mathbb{R}\ \exists y\in\mathbb{R}\text{ such that } y>x+336.$$
(a) Give the symbolic negation step-by-step.
(b) Translate both the original statement and its negation into plain English to confirm th... | [
{
"method_name": "English Semantics Cross-Check",
"approach": "Translate to English, negate carefully, then translate back to symbols.",
"steps": [
"Step 1: Original: 'For every real $x$, there exists a real $y$ that is greater than $x+336$.'",
"Step 2: Negation: 'There exists a real $x$ suc... | {
"consistency_check": "Cross-check: both derivations land on the same invariant quantity. Final answer: $\\boxed{\\exists x\\in\\mathbb{R}$.\nThe formal quantifier-negation rules and the English semantics yield the same statement: the existence of an $x$ that upper-bounds all reals after shifting by a constant.",
... | [
{
"error_description": "Negated only the inequality but kept quantifiers unchanged.",
"why_plausible": "The inequality is the most visible part of the statement, so attention focuses there.",
"why_wrong": "Quantifiers determine scope; the correct negation flips $\\forall\\leftrightarrow\\exists$ and cha... | Core principle: Negating quantified statements requires flipping each quantifier and negating the predicate while keeping variable order. Always sanity-check by translating to plain English. (Here the result is $\boxed{\exists x\in\mathbb{R}$.) |
math-001969 | Logic: Quantifiers — Negation Rules | 1 | Challenge: Negate the statement and simplify (push negation inward) using quantifier laws:
$$\forall x\in\mathbb{R}\ \exists y\in\mathbb{R}\text{ such that } y>x+75.$$
(a) Give the symbolic negation step-by-step.
(b) Translate both the original statement and its negation into plain English to confirm the meaning.
Inc... | [
{
"method_name": "English Semantics Cross-Check",
"approach": "Translate to English, negate carefully, then translate back to symbols.",
"steps": [
"Step 1: Original: 'For every real $x$, there exists a real $y$ that is greater than $x+75$.'",
"Step 2: Negation: 'There exists a real $x$ such... | {
"consistency_check": "Cross-check: both derivations land on the same invariant quantity. Final answer: $\\boxed{\\exists x\\in\\mathbb{R}$.\nThe formal quantifier-negation rules and the English semantics yield the same statement: the existence of an $x$ that upper-bounds all reals after shifting by a constant.",
... | [
{
"error_description": "Negated only the inequality but kept quantifiers unchanged.",
"why_plausible": "The inequality is the most visible part of the statement, so attention focuses there.",
"why_wrong": "Quantifiers determine scope; the correct negation flips $\\forall\\leftrightarrow\\exists$ and cha... | Key idea: Negating quantified statements requires flipping each quantifier and negating the predicate while keeping variable order. Always sanity-check by translating to plain English. (Here the result is $\boxed{\exists x\in\mathbb{R}$.) |
math-001970 | Logic: Scope — Order of Quantifiers | 1 | Show all reasoning: Negate the statement and simplify (push negation inward) using quantifier laws:
$$\forall x\in\mathbb{R}\ \exists y\in\mathbb{R}\text{ such that } y>x+384.$$
(a) Give the symbolic negation step-by-step.
(b) Translate both the original statement and its negation into plain English to confirm the mea... | [
{
"method_name": "English Semantics Cross-Check",
"approach": "Translate to English, negate carefully, then translate back to symbols.",
"steps": [
"Step 1: Original: 'For every real $x$, there exists a real $y$ that is greater than $x+384$.'",
"Step 2: Negation: 'There exists a real $x$ suc... | {
"consistency_check": "The two methods are consistent and must coincide. Final answer: $\\boxed{\\exists x\\in\\mathbb{R}$.\nThe formal quantifier-negation rules and the English semantics yield the same statement: the existence of an $x$ that upper-bounds all reals after shifting by a constant.",
"robustness_analy... | [
{
"error_description": "Negated only the inequality but kept quantifiers unchanged.",
"why_plausible": "The inequality is the most visible part of the statement, so attention focuses there.",
"why_wrong": "Quantifiers determine scope; the correct negation flips $\\forall\\leftrightarrow\\exists$ and cha... | Remember: Negating quantified statements requires flipping each quantifier and negating the predicate while keeping variable order. Always sanity-check by translating to plain English. (Here the result is $\boxed{\exists x\in\mathbb{R}$.) |
math-001971 | Logic: Scope — Order of Quantifiers | 1 | Problem: Negate the statement and simplify (push negation inward) using quantifier laws:
$$\forall x\in\mathbb{R}\ \exists y\in\mathbb{R}\text{ such that } y>x+106.$$
(a) Give the symbolic negation step-by-step.
(b) Translate both the original statement and its negation into plain English to confirm the meaning.
Incl... | [
{
"method_name": "English Semantics Cross-Check",
"approach": "Translate to English, negate carefully, then translate back to symbols.",
"steps": [
"Step 1: Original: 'For every real $x$, there exists a real $y$ that is greater than $x+106$.'",
"Step 2: Negation: 'There exists a real $x$ suc... | {
"consistency_check": "Both approaches agree after simplification. Final answer: $\\boxed{\\exists x\\in\\mathbb{R}$.\nThe formal quantifier-negation rules and the English semantics yield the same statement: the existence of an $x$ that upper-bounds all reals after shifting by a constant.",
"robustness_analysis": ... | [
{
"error_description": "Negated only the inequality but kept quantifiers unchanged.",
"why_plausible": "The inequality is the most visible part of the statement, so attention focuses there.",
"why_wrong": "Quantifiers determine scope; the correct negation flips $\\forall\\leftrightarrow\\exists$ and cha... | Core principle: Negating quantified statements requires flipping each quantifier and negating the predicate while keeping variable order. Always sanity-check by translating to plain English. (Here the result is $\boxed{\exists x\in\mathbb{R}$.) |
math-001972 | Logic: Quantifiers — Negation Rules | 1 | Warm-up: Negate the statement and simplify (push negation inward) using quantifier laws:
$$\forall x\in\mathbb{R}\ \exists y\in\mathbb{R}\text{ such that } y>x+40.$$
(a) Give the symbolic negation step-by-step.
(b) Translate both the original statement and its negation into plain English to confirm the meaning.
Inclu... | [
{
"method_name": "Quantifier Laws",
"approach": "Use $\\neg\\forall=\\exists\\neg$ and $\\neg\\exists=\\forall\\neg$, then negate the inequality.",
"steps": [
"Step 1: Start: $\\neg(\\forall x\\ \\exists y\\ P(x,y))$.",
"Step 2: Apply $\\neg\\forall x\\ Q(x)\\equiv \\exists x\\ \\neg Q(x)$ t... | {
"consistency_check": "Consistency verification shows both paths yield the identical boxed result. Final answer: $\\boxed{\\exists x\\in\\mathbb{R}$.\nThe formal quantifier-negation rules and the English semantics yield the same statement: the existence of an $x$ that upper-bounds all reals after shifting by a const... | [
{
"error_description": "Negated only the inequality but kept quantifiers unchanged.",
"why_plausible": "The inequality is the most visible part of the statement, so attention focuses there.",
"why_wrong": "Quantifiers determine scope; the correct negation flips $\\forall\\leftrightarrow\\exists$ and cha... | Remember: Negating quantified statements requires flipping each quantifier and negating the predicate while keeping variable order. Always sanity-check by translating to plain English. (Here the result is $\boxed{\exists x\in\mathbb{R}$.) |
math-001973 | Logic: Scope — Order of Quantifiers | 1 | Keep the final answer in boxed form: Negate the statement and simplify (push negation inward) using quantifier laws:
$$\forall x\in\mathbb{R}\ \exists y\in\mathbb{R}\text{ such that } y>x+362.$$
(a) Give the symbolic negation step-by-step.
(b) Translate both the original statement and its negation into plain English t... | [
{
"method_name": "Quantifier Laws",
"approach": "Use $\\neg\\forall=\\exists\\neg$ and $\\neg\\exists=\\forall\\neg$, then negate the inequality.",
"steps": [
"Step 1: Start: $\\neg(\\forall x\\ \\exists y\\ P(x,y))$.",
"Step 2: Apply $\\neg\\forall x\\ Q(x)\\equiv \\exists x\\ \\neg Q(x)$ t... | {
"consistency_check": "Both approaches agree after simplification. Final answer: $\\boxed{\\exists x\\in\\mathbb{R}$.\nThe formal quantifier-negation rules and the English semantics yield the same statement: the existence of an $x$ that upper-bounds all reals after shifting by a constant.",
"robustness_analysis": ... | [
{
"error_description": "Negated only the inequality but kept quantifiers unchanged.",
"why_plausible": "The inequality is the most visible part of the statement, so attention focuses there.",
"why_wrong": "Quantifiers determine scope; the correct negation flips $\\forall\\leftrightarrow\\exists$ and cha... | Core principle: Negating quantified statements requires flipping each quantifier and negating the predicate while keeping variable order. Always sanity-check by translating to plain English. |
math-001974 | Foundations: Translating Statements | 1 | Work this out carefully: Negate the statement and simplify (push negation inward) using quantifier laws:
$$\forall x\in\mathbb{R}\ \exists y\in\mathbb{R}\text{ such that } y>x+342.$$
(a) Give the symbolic negation step-by-step.
(b) Translate both the original statement and its negation into plain English to confirm th... | [
{
"method_name": "English Semantics Cross-Check",
"approach": "Translate to English, negate carefully, then translate back to symbols.",
"steps": [
"Step 1: Original: 'For every real $x$, there exists a real $y$ that is greater than $x+342$.'",
"Step 2: Negation: 'There exists a real $x$ suc... | {
"consistency_check": "Consistency verification shows both paths yield the identical boxed result. Final answer: $\\boxed{\\exists x\\in\\mathbb{R}$.\nThe formal quantifier-negation rules and the English semantics yield the same statement: the existence of an $x$ that upper-bounds all reals after shifting by a const... | [
{
"error_description": "Negated only the inequality but kept quantifiers unchanged.",
"why_plausible": "The inequality is the most visible part of the statement, so attention focuses there.",
"why_wrong": "Quantifiers determine scope; the correct negation flips $\\forall\\leftrightarrow\\exists$ and cha... | Remember: Negating quantified statements requires flipping each quantifier and negating the predicate while keeping variable order. Always sanity-check by translating to plain English. |
math-001975 | Logic: Quantifiers — Negation Rules | 1 | Provide both a computational and a conceptual explanation: Negate the statement and simplify (push negation inward) using quantifier laws:
$$\forall x\in\mathbb{R}\ \exists y\in\mathbb{R}\text{ such that } y>x+251.$$
(a) Give the symbolic negation step-by-step.
(b) Translate both the original statement and its negatio... | [
{
"method_name": "Quantifier Laws",
"approach": "Use $\\neg\\forall=\\exists\\neg$ and $\\neg\\exists=\\forall\\neg$, then negate the inequality.",
"steps": [
"Step 1: Start: $\\neg(\\forall x\\ \\exists y\\ P(x,y))$.",
"Step 2: Apply $\\neg\\forall x\\ Q(x)\\equiv \\exists x\\ \\neg Q(x)$ t... | {
"consistency_check": "The two methods are consistent and must coincide. Final answer: $\\boxed{\\exists x\\in\\mathbb{R}$.\nThe formal quantifier-negation rules and the English semantics yield the same statement: the existence of an $x$ that upper-bounds all reals after shifting by a constant.",
"robustness_analy... | [
{
"error_description": "Negated only the inequality but kept quantifiers unchanged.",
"why_plausible": "The inequality is the most visible part of the statement, so attention focuses there.",
"why_wrong": "Quantifiers determine scope; the correct negation flips $\\forall\\leftrightarrow\\exists$ and cha... | Takeaway: Negating quantified statements requires flipping each quantifier and negating the predicate while keeping variable order. Always sanity-check by translating to plain English. |
math-001976 | Foundations: Translating Statements | 1 | Solve and include a self-check: Negate the statement and simplify (push negation inward) using quantifier laws:
$$\forall x\in\mathbb{R}\ \exists y\in\mathbb{R}\text{ such that } y>x+220.$$
(a) Give the symbolic negation step-by-step.
(b) Translate both the original statement and its negation into plain English to con... | [
{
"method_name": "English Semantics Cross-Check",
"approach": "Translate to English, negate carefully, then translate back to symbols.",
"steps": [
"Step 1: Original: 'For every real $x$, there exists a real $y$ that is greater than $x+220$.'",
"Step 2: Negation: 'There exists a real $x$ suc... | {
"consistency_check": "Cross-check: both derivations land on the same invariant quantity. Final answer: $\\boxed{\\exists x\\in\\mathbb{R}$.\nThe formal quantifier-negation rules and the English semantics yield the same statement: the existence of an $x$ that upper-bounds all reals after shifting by a constant.",
... | [
{
"error_description": "Negated only the inequality but kept quantifiers unchanged.",
"why_plausible": "The inequality is the most visible part of the statement, so attention focuses there.",
"why_wrong": "Quantifiers determine scope; the correct negation flips $\\forall\\leftrightarrow\\exists$ and cha... | Core principle: Negating quantified statements requires flipping each quantifier and negating the predicate while keeping variable order. Always sanity-check by translating to plain English. (Here the result is $\boxed{\exists x\in\mathbb{R}$.) |
math-001977 | Logic: Formal vs English Paraphrase | 1 | Solve and sanity-check: Negate the statement and simplify (push negation inward) using quantifier laws:
$$\forall x\in\mathbb{R}\ \exists y\in\mathbb{R}\text{ such that } y>x+190.$$
(a) Give the symbolic negation step-by-step.
(b) Translate both the original statement and its negation into plain English to confirm the... | [
{
"method_name": "Quantifier Laws",
"approach": "Use $\\neg\\forall=\\exists\\neg$ and $\\neg\\exists=\\forall\\neg$, then negate the inequality.",
"steps": [
"Step 1: Start: $\\neg(\\forall x\\ \\exists y\\ P(x,y))$.",
"Step 2: Apply $\\neg\\forall x\\ Q(x)\\equiv \\exists x\\ \\neg Q(x)$ t... | {
"consistency_check": "The two methods are consistent and must coincide. Final answer: $\\boxed{\\exists x\\in\\mathbb{R}$.\nThe formal quantifier-negation rules and the English semantics yield the same statement: the existence of an $x$ that upper-bounds all reals after shifting by a constant.",
"robustness_analy... | [
{
"error_description": "Negated only the inequality but kept quantifiers unchanged.",
"why_plausible": "The inequality is the most visible part of the statement, so attention focuses there.",
"why_wrong": "Quantifiers determine scope; the correct negation flips $\\forall\\leftrightarrow\\exists$ and cha... | Key idea: Negating quantified statements requires flipping each quantifier and negating the predicate while keeping variable order. Always sanity-check by translating to plain English. |
math-001978 | Logic: Scope — Order of Quantifiers | 1 | Solve (and briefly cross-validate): Negate the statement and simplify (push negation inward) using quantifier laws:
$$\forall x\in\mathbb{R}\ \exists y\in\mathbb{R}\text{ such that } y>x+367.$$
(a) Give the symbolic negation step-by-step.
(b) Translate both the original statement and its negation into plain English to... | [
{
"method_name": "English Semantics Cross-Check",
"approach": "Translate to English, negate carefully, then translate back to symbols.",
"steps": [
"Step 1: Original: 'For every real $x$, there exists a real $y$ that is greater than $x+367$.'",
"Step 2: Negation: 'There exists a real $x$ suc... | {
"consistency_check": "Consistency verification shows both paths yield the identical boxed result. Final answer: $\\boxed{\\exists x\\in\\mathbb{R}$.\nThe formal quantifier-negation rules and the English semantics yield the same statement: the existence of an $x$ that upper-bounds all reals after shifting by a const... | [
{
"error_description": "Negated only the inequality but kept quantifiers unchanged.",
"why_plausible": "The inequality is the most visible part of the statement, so attention focuses there.",
"why_wrong": "Quantifiers determine scope; the correct negation flips $\\forall\\leftrightarrow\\exists$ and cha... | Key idea: Negating quantified statements requires flipping each quantifier and negating the predicate while keeping variable order. Always sanity-check by translating to plain English. (Here the result is $\boxed{\exists x\in\mathbb{R}$.) |
math-001979 | Foundations: Translating Statements | 1 | Complete the analysis: Negate the statement and simplify (push negation inward) using quantifier laws:
$$\forall x\in\mathbb{R}\ \exists y\in\mathbb{R}\text{ such that } y>x+140.$$
(a) Give the symbolic negation step-by-step.
(b) Translate both the original statement and its negation into plain English to confirm the ... | [
{
"method_name": "Quantifier Laws",
"approach": "Use $\\neg\\forall=\\exists\\neg$ and $\\neg\\exists=\\forall\\neg$, then negate the inequality.",
"steps": [
"Step 1: Start: $\\neg(\\forall x\\ \\exists y\\ P(x,y))$.",
"Step 2: Apply $\\neg\\forall x\\ Q(x)\\equiv \\exists x\\ \\neg Q(x)$ t... | {
"consistency_check": "The two methods are consistent and must coincide. Final answer: $\\boxed{\\exists x\\in\\mathbb{R}$.\nThe formal quantifier-negation rules and the English semantics yield the same statement: the existence of an $x$ that upper-bounds all reals after shifting by a constant.",
"robustness_analy... | [
{
"error_description": "Negated only the inequality but kept quantifiers unchanged.",
"why_plausible": "The inequality is the most visible part of the statement, so attention focuses there.",
"why_wrong": "Quantifiers determine scope; the correct negation flips $\\forall\\leftrightarrow\\exists$ and cha... | Key idea: Negating quantified statements requires flipping each quantifier and negating the predicate while keeping variable order. Always sanity-check by translating to plain English. (Here the result is $\boxed{\exists x\in\mathbb{R}$.) |
math-001980 | Foundations: Translating Statements | 1 | Make each step logically reversible (or explain if not): Negate the statement and simplify (push negation inward) using quantifier laws:
$$\forall x\in\mathbb{R}\ \exists y\in\mathbb{R}\text{ such that } y>x+212.$$
(a) Give the symbolic negation step-by-step.
(b) Translate both the original statement and its negation ... | [
{
"method_name": "Quantifier Laws",
"approach": "Use $\\neg\\forall=\\exists\\neg$ and $\\neg\\exists=\\forall\\neg$, then negate the inequality.",
"steps": [
"Step 1: Start: $\\neg(\\forall x\\ \\exists y\\ P(x,y))$.",
"Step 2: Apply $\\neg\\forall x\\ Q(x)\\equiv \\exists x\\ \\neg Q(x)$ t... | {
"consistency_check": "Cross-check: both derivations land on the same invariant quantity. Final answer: $\\boxed{\\exists x\\in\\mathbb{R}$.\nThe formal quantifier-negation rules and the English semantics yield the same statement: the existence of an $x$ that upper-bounds all reals after shifting by a constant.",
... | [
{
"error_description": "Negated only the inequality but kept quantifiers unchanged.",
"why_plausible": "The inequality is the most visible part of the statement, so attention focuses there.",
"why_wrong": "Quantifiers determine scope; the correct negation flips $\\forall\\leftrightarrow\\exists$ and cha... | Key idea: Negating quantified statements requires flipping each quantifier and negating the predicate while keeping variable order. Always sanity-check by translating to plain English. |
math-001981 | Logic: Quantifiers — Negation Rules | 1 | Work this out carefully: Negate the statement and simplify (push negation inward) using quantifier laws:
$$\forall x\in\mathbb{R}\ \exists y\in\mathbb{R}\text{ such that } y>x+91.$$
(a) Give the symbolic negation step-by-step.
(b) Translate both the original statement and its negation into plain English to confirm the... | [
{
"method_name": "English Semantics Cross-Check",
"approach": "Translate to English, negate carefully, then translate back to symbols.",
"steps": [
"Step 1: Original: 'For every real $x$, there exists a real $y$ that is greater than $x+91$.'",
"Step 2: Negation: 'There exists a real $x$ such... | {
"consistency_check": "Consistency verification shows both paths yield the identical boxed result. Final answer: $\\boxed{\\exists x\\in\\mathbb{R}$.\nThe formal quantifier-negation rules and the English semantics yield the same statement: the existence of an $x$ that upper-bounds all reals after shifting by a const... | [
{
"error_description": "Negated only the inequality but kept quantifiers unchanged.",
"why_plausible": "The inequality is the most visible part of the statement, so attention focuses there.",
"why_wrong": "Quantifiers determine scope; the correct negation flips $\\forall\\leftrightarrow\\exists$ and cha... | Takeaway: Negating quantified statements requires flipping each quantifier and negating the predicate while keeping variable order. Always sanity-check by translating to plain English. (Here the result is $\boxed{\exists x\in\mathbb{R}$.) |
math-001982 | Logic: Formal vs English Paraphrase | 1 | Solve with verification: Negate the statement and simplify (push negation inward) using quantifier laws:
$$\forall x\in\mathbb{R}\ \exists y\in\mathbb{R}\text{ such that } y>x+32.$$
(a) Give the symbolic negation step-by-step.
(b) Translate both the original statement and its negation into plain English to confirm the... | [
{
"method_name": "English Semantics Cross-Check",
"approach": "Translate to English, negate carefully, then translate back to symbols.",
"steps": [
"Step 1: Original: 'For every real $x$, there exists a real $y$ that is greater than $x+32$.'",
"Step 2: Negation: 'There exists a real $x$ such... | {
"consistency_check": "Both approaches agree after simplification. Final answer: $\\boxed{\\exists x\\in\\mathbb{R}$.\nThe formal quantifier-negation rules and the English semantics yield the same statement: the existence of an $x$ that upper-bounds all reals after shifting by a constant.",
"robustness_analysis": ... | [
{
"error_description": "Negated only the inequality but kept quantifiers unchanged.",
"why_plausible": "The inequality is the most visible part of the statement, so attention focuses there.",
"why_wrong": "Quantifiers determine scope; the correct negation flips $\\forall\\leftrightarrow\\exists$ and cha... | Remember: Negating quantified statements requires flipping each quantifier and negating the predicate while keeping variable order. Always sanity-check by translating to plain English. |
math-001983 | Logic: Formal vs English Paraphrase | 1 | Solve and sanity-check: Negate the statement and simplify (push negation inward) using quantifier laws:
$$\forall x\in\mathbb{R}\ \exists y\in\mathbb{R}\text{ such that } y>x+93.$$
(a) Give the symbolic negation step-by-step.
(b) Translate both the original statement and its negation into plain English to confirm the ... | [
{
"method_name": "Quantifier Laws",
"approach": "Use $\\neg\\forall=\\exists\\neg$ and $\\neg\\exists=\\forall\\neg$, then negate the inequality.",
"steps": [
"Step 1: Start: $\\neg(\\forall x\\ \\exists y\\ P(x,y))$.",
"Step 2: Apply $\\neg\\forall x\\ Q(x)\\equiv \\exists x\\ \\neg Q(x)$ t... | {
"consistency_check": "The two methods are consistent and must coincide. Final answer: $\\boxed{\\exists x\\in\\mathbb{R}$.\nThe formal quantifier-negation rules and the English semantics yield the same statement: the existence of an $x$ that upper-bounds all reals after shifting by a constant.",
"robustness_analy... | [
{
"error_description": "Negated only the inequality but kept quantifiers unchanged.",
"why_plausible": "The inequality is the most visible part of the statement, so attention focuses there.",
"why_wrong": "Quantifiers determine scope; the correct negation flips $\\forall\\leftrightarrow\\exists$ and cha... | Core principle: Negating quantified statements requires flipping each quantifier and negating the predicate while keeping variable order. Always sanity-check by translating to plain English. (Here the result is $\boxed{\exists x\in\mathbb{R}$.) |
math-001984 | Logic: Formal vs English Paraphrase | 1 | Write the solution set clearly: Negate the statement and simplify (push negation inward) using quantifier laws:
$$\forall x\in\mathbb{R}\ \exists y\in\mathbb{R}\text{ such that } y>x+39.$$
(a) Give the symbolic negation step-by-step.
(b) Translate both the original statement and its negation into plain English to conf... | [
{
"method_name": "English Semantics Cross-Check",
"approach": "Translate to English, negate carefully, then translate back to symbols.",
"steps": [
"Step 1: Original: 'For every real $x$, there exists a real $y$ that is greater than $x+39$.'",
"Step 2: Negation: 'There exists a real $x$ such... | {
"consistency_check": "Cross-check: both derivations land on the same invariant quantity. Final answer: $\\boxed{\\exists x\\in\\mathbb{R}$.\nThe formal quantifier-negation rules and the English semantics yield the same statement: the existence of an $x$ that upper-bounds all reals after shifting by a constant.",
... | [
{
"error_description": "Negated only the inequality but kept quantifiers unchanged.",
"why_plausible": "The inequality is the most visible part of the statement, so attention focuses there.",
"why_wrong": "Quantifiers determine scope; the correct negation flips $\\forall\\leftrightarrow\\exists$ and cha... | Takeaway: Negating quantified statements requires flipping each quantifier and negating the predicate while keeping variable order. Always sanity-check by translating to plain English. |
math-001985 | Foundations: Translating Statements | 1 | Give a fully justified solution: Negate the statement and simplify (push negation inward) using quantifier laws:
$$\forall x\in\mathbb{R}\ \exists y\in\mathbb{R}\text{ such that } y>x+400.$$
(a) Give the symbolic negation step-by-step.
(b) Translate both the original statement and its negation into plain English to co... | [
{
"method_name": "English Semantics Cross-Check",
"approach": "Translate to English, negate carefully, then translate back to symbols.",
"steps": [
"Step 1: Original: 'For every real $x$, there exists a real $y$ that is greater than $x+400$.'",
"Step 2: Negation: 'There exists a real $x$ suc... | {
"consistency_check": "Both approaches agree after simplification. Final answer: $\\boxed{\\exists x\\in\\mathbb{R}$.\nThe formal quantifier-negation rules and the English semantics yield the same statement: the existence of an $x$ that upper-bounds all reals after shifting by a constant.",
"robustness_analysis": ... | [
{
"error_description": "Negated only the inequality but kept quantifiers unchanged.",
"why_plausible": "The inequality is the most visible part of the statement, so attention focuses there.",
"why_wrong": "Quantifiers determine scope; the correct negation flips $\\forall\\leftrightarrow\\exists$ and cha... | Takeaway: Negating quantified statements requires flipping each quantifier and negating the predicate while keeping variable order. Always sanity-check by translating to plain English. |
math-001986 | Logic: Formal vs English Paraphrase | 1 | Problem: Negate the statement and simplify (push negation inward) using quantifier laws:
$$\forall x\in\mathbb{R}\ \exists y\in\mathbb{R}\text{ such that } y>x+213.$$
(a) Give the symbolic negation step-by-step.
(b) Translate both the original statement and its negation into plain English to confirm the meaning.
Incl... | [
{
"method_name": "Quantifier Laws",
"approach": "Use $\\neg\\forall=\\exists\\neg$ and $\\neg\\exists=\\forall\\neg$, then negate the inequality.",
"steps": [
"Step 1: Start: $\\neg(\\forall x\\ \\exists y\\ P(x,y))$.",
"Step 2: Apply $\\neg\\forall x\\ Q(x)\\equiv \\exists x\\ \\neg Q(x)$ t... | {
"consistency_check": "Cross-check: both derivations land on the same invariant quantity. Final answer: $\\boxed{\\exists x\\in\\mathbb{R}$.\nThe formal quantifier-negation rules and the English semantics yield the same statement: the existence of an $x$ that upper-bounds all reals after shifting by a constant.",
... | [
{
"error_description": "Negated only the inequality but kept quantifiers unchanged.",
"why_plausible": "The inequality is the most visible part of the statement, so attention focuses there.",
"why_wrong": "Quantifiers determine scope; the correct negation flips $\\forall\\leftrightarrow\\exists$ and cha... | Key idea: Negating quantified statements requires flipping each quantifier and negating the predicate while keeping variable order. Always sanity-check by translating to plain English. (Here the result is $\boxed{\exists x\in\mathbb{R}$.) |
math-001987 | Logic: Quantifiers — Negation Rules | 1 | Compute the requested quantity: Negate the statement and simplify (push negation inward) using quantifier laws:
$$\forall x\in\mathbb{R}\ \exists y\in\mathbb{R}\text{ such that } y>x+314.$$
(a) Give the symbolic negation step-by-step.
(b) Translate both the original statement and its negation into plain English to con... | [
{
"method_name": "Quantifier Laws",
"approach": "Use $\\neg\\forall=\\exists\\neg$ and $\\neg\\exists=\\forall\\neg$, then negate the inequality.",
"steps": [
"Step 1: Start: $\\neg(\\forall x\\ \\exists y\\ P(x,y))$.",
"Step 2: Apply $\\neg\\forall x\\ Q(x)\\equiv \\exists x\\ \\neg Q(x)$ t... | {
"consistency_check": "Consistency verification shows both paths yield the identical boxed result. Final answer: $\\boxed{\\exists x\\in\\mathbb{R}$.\nThe formal quantifier-negation rules and the English semantics yield the same statement: the existence of an $x$ that upper-bounds all reals after shifting by a const... | [
{
"error_description": "Negated only the inequality but kept quantifiers unchanged.",
"why_plausible": "The inequality is the most visible part of the statement, so attention focuses there.",
"why_wrong": "Quantifiers determine scope; the correct negation flips $\\forall\\leftrightarrow\\exists$ and cha... | Takeaway: Negating quantified statements requires flipping each quantifier and negating the predicate while keeping variable order. Always sanity-check by translating to plain English. (Here the result is $\boxed{\exists x\in\mathbb{R}$.) |
math-001988 | Logic: Scope — Order of Quantifiers | 1 | Find the exact value: Negate the statement and simplify (push negation inward) using quantifier laws:
$$\forall x\in\mathbb{R}\ \exists y\in\mathbb{R}\text{ such that } y>x+178.$$
(a) Give the symbolic negation step-by-step.
(b) Translate both the original statement and its negation into plain English to confirm the m... | [
{
"method_name": "English Semantics Cross-Check",
"approach": "Translate to English, negate carefully, then translate back to symbols.",
"steps": [
"Step 1: Original: 'For every real $x$, there exists a real $y$ that is greater than $x+178$.'",
"Step 2: Negation: 'There exists a real $x$ suc... | {
"consistency_check": "Both approaches agree after simplification. Final answer: $\\boxed{\\exists x\\in\\mathbb{R}$.\nThe formal quantifier-negation rules and the English semantics yield the same statement: the existence of an $x$ that upper-bounds all reals after shifting by a constant.",
"robustness_analysis": ... | [
{
"error_description": "Negated only the inequality but kept quantifiers unchanged.",
"why_plausible": "The inequality is the most visible part of the statement, so attention focuses there.",
"why_wrong": "Quantifiers determine scope; the correct negation flips $\\forall\\leftrightarrow\\exists$ and cha... | Key idea: Negating quantified statements requires flipping each quantifier and negating the predicate while keeping variable order. Always sanity-check by translating to plain English. (Here the result is $\boxed{\exists x\in\mathbb{R}$.) |
math-001989 | Foundations: Translating Statements | 1 | Give reasoning, not just computation: Negate the statement and simplify (push negation inward) using quantifier laws:
$$\forall x\in\mathbb{R}\ \exists y\in\mathbb{R}\text{ such that } y>x+122.$$
(a) Give the symbolic negation step-by-step.
(b) Translate both the original statement and its negation into plain English ... | [
{
"method_name": "English Semantics Cross-Check",
"approach": "Translate to English, negate carefully, then translate back to symbols.",
"steps": [
"Step 1: Original: 'For every real $x$, there exists a real $y$ that is greater than $x+122$.'",
"Step 2: Negation: 'There exists a real $x$ suc... | {
"consistency_check": "Both approaches agree after simplification. Final answer: $\\boxed{\\exists x\\in\\mathbb{R}$.\nThe formal quantifier-negation rules and the English semantics yield the same statement: the existence of an $x$ that upper-bounds all reals after shifting by a constant.",
"robustness_analysis": ... | [
{
"error_description": "Negated only the inequality but kept quantifiers unchanged.",
"why_plausible": "The inequality is the most visible part of the statement, so attention focuses there.",
"why_wrong": "Quantifiers determine scope; the correct negation flips $\\forall\\leftrightarrow\\exists$ and cha... | Takeaway: Negating quantified statements requires flipping each quantifier and negating the predicate while keeping variable order. Always sanity-check by translating to plain English. (Here the result is $\boxed{\exists x\in\mathbb{R}$.) |
math-001990 | Logic: Quantifiers — Negation Rules | 1 | Compute the requested quantity: Negate the statement and simplify (push negation inward) using quantifier laws:
$$\forall x\in\mathbb{R}\ \exists y\in\mathbb{R}\text{ such that } y>x+320.$$
(a) Give the symbolic negation step-by-step.
(b) Translate both the original statement and its negation into plain English to con... | [
{
"method_name": "Quantifier Laws",
"approach": "Use $\\neg\\forall=\\exists\\neg$ and $\\neg\\exists=\\forall\\neg$, then negate the inequality.",
"steps": [
"Step 1: Start: $\\neg(\\forall x\\ \\exists y\\ P(x,y))$.",
"Step 2: Apply $\\neg\\forall x\\ Q(x)\\equiv \\exists x\\ \\neg Q(x)$ t... | {
"consistency_check": "The two methods are consistent and must coincide. Final answer: $\\boxed{\\exists x\\in\\mathbb{R}$.\nThe formal quantifier-negation rules and the English semantics yield the same statement: the existence of an $x$ that upper-bounds all reals after shifting by a constant.",
"robustness_analy... | [
{
"error_description": "Negated only the inequality but kept quantifiers unchanged.",
"why_plausible": "The inequality is the most visible part of the statement, so attention focuses there.",
"why_wrong": "Quantifiers determine scope; the correct negation flips $\\forall\\leftrightarrow\\exists$ and cha... | Remember: Negating quantified statements requires flipping each quantifier and negating the predicate while keeping variable order. Always sanity-check by translating to plain English. |
math-001991 | Logic: Quantifiers — Negation Rules | 1 | Write the solution set clearly: Negate the statement and simplify (push negation inward) using quantifier laws:
$$\forall x\in\mathbb{R}\ \exists y\in\mathbb{R}\text{ such that } y>x+112.$$
(a) Give the symbolic negation step-by-step.
(b) Translate both the original statement and its negation into plain English to con... | [
{
"method_name": "Quantifier Laws",
"approach": "Use $\\neg\\forall=\\exists\\neg$ and $\\neg\\exists=\\forall\\neg$, then negate the inequality.",
"steps": [
"Step 1: Start: $\\neg(\\forall x\\ \\exists y\\ P(x,y))$.",
"Step 2: Apply $\\neg\\forall x\\ Q(x)\\equiv \\exists x\\ \\neg Q(x)$ t... | {
"consistency_check": "Cross-check: both derivations land on the same invariant quantity. Final answer: $\\boxed{\\exists x\\in\\mathbb{R}$.\nThe formal quantifier-negation rules and the English semantics yield the same statement: the existence of an $x$ that upper-bounds all reals after shifting by a constant.",
... | [
{
"error_description": "Negated only the inequality but kept quantifiers unchanged.",
"why_plausible": "The inequality is the most visible part of the statement, so attention focuses there.",
"why_wrong": "Quantifiers determine scope; the correct negation flips $\\forall\\leftrightarrow\\exists$ and cha... | Key idea: Negating quantified statements requires flipping each quantifier and negating the predicate while keeping variable order. Always sanity-check by translating to plain English. (Here the result is $\boxed{\exists x\in\mathbb{R}$.) |
math-001992 | Logic: Scope — Order of Quantifiers | 1 | Solve and include a self-check: Negate the statement and simplify (push negation inward) using quantifier laws:
$$\forall x\in\mathbb{R}\ \exists y\in\mathbb{R}\text{ such that } y>x+331.$$
(a) Give the symbolic negation step-by-step.
(b) Translate both the original statement and its negation into plain English to con... | [
{
"method_name": "Quantifier Laws",
"approach": "Use $\\neg\\forall=\\exists\\neg$ and $\\neg\\exists=\\forall\\neg$, then negate the inequality.",
"steps": [
"Step 1: Start: $\\neg(\\forall x\\ \\exists y\\ P(x,y))$.",
"Step 2: Apply $\\neg\\forall x\\ Q(x)\\equiv \\exists x\\ \\neg Q(x)$ t... | {
"consistency_check": "Cross-check: both derivations land on the same invariant quantity. Final answer: $\\boxed{\\exists x\\in\\mathbb{R}$.\nThe formal quantifier-negation rules and the English semantics yield the same statement: the existence of an $x$ that upper-bounds all reals after shifting by a constant.",
... | [
{
"error_description": "Negated only the inequality but kept quantifiers unchanged.",
"why_plausible": "The inequality is the most visible part of the statement, so attention focuses there.",
"why_wrong": "Quantifiers determine scope; the correct negation flips $\\forall\\leftrightarrow\\exists$ and cha... | Remember: Negating quantified statements requires flipping each quantifier and negating the predicate while keeping variable order. Always sanity-check by translating to plain English. |
math-001993 | Logic: Scope — Order of Quantifiers | 1 | Track quantifiers carefully: Negate the statement and simplify (push negation inward) using quantifier laws:
$$\forall x\in\mathbb{R}\ \exists y\in\mathbb{R}\text{ such that } y>x+50.$$
(a) Give the symbolic negation step-by-step.
(b) Translate both the original statement and its negation into plain English to confirm... | [
{
"method_name": "English Semantics Cross-Check",
"approach": "Translate to English, negate carefully, then translate back to symbols.",
"steps": [
"Step 1: Original: 'For every real $x$, there exists a real $y$ that is greater than $x+50$.'",
"Step 2: Negation: 'There exists a real $x$ such... | {
"consistency_check": "Both approaches agree after simplification. Final answer: $\\boxed{\\exists x\\in\\mathbb{R}$.\nThe formal quantifier-negation rules and the English semantics yield the same statement: the existence of an $x$ that upper-bounds all reals after shifting by a constant.",
"robustness_analysis": ... | [
{
"error_description": "Negated only the inequality but kept quantifiers unchanged.",
"why_plausible": "The inequality is the most visible part of the statement, so attention focuses there.",
"why_wrong": "Quantifiers determine scope; the correct negation flips $\\forall\\leftrightarrow\\exists$ and cha... | Remember: Negating quantified statements requires flipping each quantifier and negating the predicate while keeping variable order. Always sanity-check by translating to plain English. (Here the result is $\boxed{\exists x\in\mathbb{R}$.) |
math-001994 | Logic: Formal vs English Paraphrase | 1 | Prompt: Negate the statement and simplify (push negation inward) using quantifier laws:
$$\forall x\in\mathbb{R}\ \exists y\in\mathbb{R}\text{ such that } y>x+383.$$
(a) Give the symbolic negation step-by-step.
(b) Translate both the original statement and its negation into plain English to confirm the meaning.
Inclu... | [
{
"method_name": "Quantifier Laws",
"approach": "Use $\\neg\\forall=\\exists\\neg$ and $\\neg\\exists=\\forall\\neg$, then negate the inequality.",
"steps": [
"Step 1: Start: $\\neg(\\forall x\\ \\exists y\\ P(x,y))$.",
"Step 2: Apply $\\neg\\forall x\\ Q(x)\\equiv \\exists x\\ \\neg Q(x)$ t... | {
"consistency_check": "Both approaches agree after simplification. Final answer: $\\boxed{\\exists x\\in\\mathbb{R}$.\nThe formal quantifier-negation rules and the English semantics yield the same statement: the existence of an $x$ that upper-bounds all reals after shifting by a constant.",
"robustness_analysis": ... | [
{
"error_description": "Negated only the inequality but kept quantifiers unchanged.",
"why_plausible": "The inequality is the most visible part of the statement, so attention focuses there.",
"why_wrong": "Quantifiers determine scope; the correct negation flips $\\forall\\leftrightarrow\\exists$ and cha... | Takeaway: Negating quantified statements requires flipping each quantifier and negating the predicate while keeping variable order. Always sanity-check by translating to plain English. |
math-001995 | Logic: Formal vs English Paraphrase | 1 | Indicate where a theorem is used: Negate the statement and simplify (push negation inward) using quantifier laws:
$$\forall x\in\mathbb{R}\ \exists y\in\mathbb{R}\text{ such that } y>x+376.$$
(a) Give the symbolic negation step-by-step.
(b) Translate both the original statement and its negation into plain English to c... | [
{
"method_name": "Quantifier Laws",
"approach": "Use $\\neg\\forall=\\exists\\neg$ and $\\neg\\exists=\\forall\\neg$, then negate the inequality.",
"steps": [
"Step 1: Start: $\\neg(\\forall x\\ \\exists y\\ P(x,y))$.",
"Step 2: Apply $\\neg\\forall x\\ Q(x)\\equiv \\exists x\\ \\neg Q(x)$ t... | {
"consistency_check": "Cross-check: both derivations land on the same invariant quantity. Final answer: $\\boxed{\\exists x\\in\\mathbb{R}$.\nThe formal quantifier-negation rules and the English semantics yield the same statement: the existence of an $x$ that upper-bounds all reals after shifting by a constant.",
... | [
{
"error_description": "Negated only the inequality but kept quantifiers unchanged.",
"why_plausible": "The inequality is the most visible part of the statement, so attention focuses there.",
"why_wrong": "Quantifiers determine scope; the correct negation flips $\\forall\\leftrightarrow\\exists$ and cha... | Remember: Negating quantified statements requires flipping each quantifier and negating the predicate while keeping variable order. Always sanity-check by translating to plain English. |
math-001996 | Logic: Scope — Order of Quantifiers | 1 | Solve with verification: Negate the statement and simplify (push negation inward) using quantifier laws:
$$\forall x\in\mathbb{R}\ \exists y\in\mathbb{R}\text{ such that } y>x+382.$$
(a) Give the symbolic negation step-by-step.
(b) Translate both the original statement and its negation into plain English to confirm th... | [
{
"method_name": "Quantifier Laws",
"approach": "Use $\\neg\\forall=\\exists\\neg$ and $\\neg\\exists=\\forall\\neg$, then negate the inequality.",
"steps": [
"Step 1: Start: $\\neg(\\forall x\\ \\exists y\\ P(x,y))$.",
"Step 2: Apply $\\neg\\forall x\\ Q(x)\\equiv \\exists x\\ \\neg Q(x)$ t... | {
"consistency_check": "Both approaches agree after simplification. Final answer: $\\boxed{\\exists x\\in\\mathbb{R}$.\nThe formal quantifier-negation rules and the English semantics yield the same statement: the existence of an $x$ that upper-bounds all reals after shifting by a constant.",
"robustness_analysis": ... | [
{
"error_description": "Negated only the inequality but kept quantifiers unchanged.",
"why_plausible": "The inequality is the most visible part of the statement, so attention focuses there.",
"why_wrong": "Quantifiers determine scope; the correct negation flips $\\forall\\leftrightarrow\\exists$ and cha... | Core principle: Negating quantified statements requires flipping each quantifier and negating the predicate while keeping variable order. Always sanity-check by translating to plain English. (Here the result is $\boxed{\exists x\in\mathbb{R}$.) |
math-001997 | Logic: Formal vs English Paraphrase | 1 | Problem: Negate the statement and simplify (push negation inward) using quantifier laws:
$$\forall x\in\mathbb{R}\ \exists y\in\mathbb{R}\text{ such that } y>x+396.$$
(a) Give the symbolic negation step-by-step.
(b) Translate both the original statement and its negation into plain English to confirm the meaning.
Incl... | [
{
"method_name": "Quantifier Laws",
"approach": "Use $\\neg\\forall=\\exists\\neg$ and $\\neg\\exists=\\forall\\neg$, then negate the inequality.",
"steps": [
"Step 1: Start: $\\neg(\\forall x\\ \\exists y\\ P(x,y))$.",
"Step 2: Apply $\\neg\\forall x\\ Q(x)\\equiv \\exists x\\ \\neg Q(x)$ t... | {
"consistency_check": "The two methods are consistent and must coincide. Final answer: $\\boxed{\\exists x\\in\\mathbb{R}$.\nThe formal quantifier-negation rules and the English semantics yield the same statement: the existence of an $x$ that upper-bounds all reals after shifting by a constant.",
"robustness_analy... | [
{
"error_description": "Negated only the inequality but kept quantifiers unchanged.",
"why_plausible": "The inequality is the most visible part of the statement, so attention focuses there.",
"why_wrong": "Quantifiers determine scope; the correct negation flips $\\forall\\leftrightarrow\\exists$ and cha... | Takeaway: Negating quantified statements requires flipping each quantifier and negating the predicate while keeping variable order. Always sanity-check by translating to plain English. (Here the result is $\boxed{\exists x\in\mathbb{R}$.) |
math-001998 | Logic: Quantifiers — Negation Rules | 1 | Give an answer and a quick verification: Negate the statement and simplify (push negation inward) using quantifier laws:
$$\forall x\in\mathbb{R}\ \exists y\in\mathbb{R}\text{ such that } y>x+308.$$
(a) Give the symbolic negation step-by-step.
(b) Translate both the original statement and its negation into plain Engli... | [
{
"method_name": "English Semantics Cross-Check",
"approach": "Translate to English, negate carefully, then translate back to symbols.",
"steps": [
"Step 1: Original: 'For every real $x$, there exists a real $y$ that is greater than $x+308$.'",
"Step 2: Negation: 'There exists a real $x$ suc... | {
"consistency_check": "Both approaches agree after simplification. Final answer: $\\boxed{\\exists x\\in\\mathbb{R}$.\nThe formal quantifier-negation rules and the English semantics yield the same statement: the existence of an $x$ that upper-bounds all reals after shifting by a constant.",
"robustness_analysis": ... | [
{
"error_description": "Negated only the inequality but kept quantifiers unchanged.",
"why_plausible": "The inequality is the most visible part of the statement, so attention focuses there.",
"why_wrong": "Quantifiers determine scope; the correct negation flips $\\forall\\leftrightarrow\\exists$ and cha... | Remember: Negating quantified statements requires flipping each quantifier and negating the predicate while keeping variable order. Always sanity-check by translating to plain English. (Here the result is $\boxed{\exists x\in\mathbb{R}$.) |
math-001999 | Logic: Quantifiers — Negation Rules | 1 | Indicate where a theorem is used: Negate the statement and simplify (push negation inward) using quantifier laws:
$$\forall x\in\mathbb{R}\ \exists y\in\mathbb{R}\text{ such that } y>x+44.$$
(a) Give the symbolic negation step-by-step.
(b) Translate both the original statement and its negation into plain English to co... | [
{
"method_name": "English Semantics Cross-Check",
"approach": "Translate to English, negate carefully, then translate back to symbols.",
"steps": [
"Step 1: Original: 'For every real $x$, there exists a real $y$ that is greater than $x+44$.'",
"Step 2: Negation: 'There exists a real $x$ such... | {
"consistency_check": "Cross-check: both derivations land on the same invariant quantity. Final answer: $\\boxed{\\exists x\\in\\mathbb{R}$.\nThe formal quantifier-negation rules and the English semantics yield the same statement: the existence of an $x$ that upper-bounds all reals after shifting by a constant.",
... | [
{
"error_description": "Negated only the inequality but kept quantifiers unchanged.",
"why_plausible": "The inequality is the most visible part of the statement, so attention focuses there.",
"why_wrong": "Quantifiers determine scope; the correct negation flips $\\forall\\leftrightarrow\\exists$ and cha... | Remember: Negating quantified statements requires flipping each quantifier and negating the predicate while keeping variable order. Always sanity-check by translating to plain English. |
math-002000 | Foundations: Translating Statements | 1 | Give an answer and a quick verification: Negate the statement and simplify (push negation inward) using quantifier laws:
$$\forall x\in\mathbb{R}\ \exists y\in\mathbb{R}\text{ such that } y>x+241.$$
(a) Give the symbolic negation step-by-step.
(b) Translate both the original statement and its negation into plain Engli... | [
{
"method_name": "Quantifier Laws",
"approach": "Use $\\neg\\forall=\\exists\\neg$ and $\\neg\\exists=\\forall\\neg$, then negate the inequality.",
"steps": [
"Step 1: Start: $\\neg(\\forall x\\ \\exists y\\ P(x,y))$.",
"Step 2: Apply $\\neg\\forall x\\ Q(x)\\equiv \\exists x\\ \\neg Q(x)$ t... | {
"consistency_check": "Cross-check: both derivations land on the same invariant quantity. Final answer: $\\boxed{\\exists x\\in\\mathbb{R}$.\nThe formal quantifier-negation rules and the English semantics yield the same statement: the existence of an $x$ that upper-bounds all reals after shifting by a constant.",
... | [
{
"error_description": "Negated only the inequality but kept quantifiers unchanged.",
"why_plausible": "The inequality is the most visible part of the statement, so attention focuses there.",
"why_wrong": "Quantifiers determine scope; the correct negation flips $\\forall\\leftrightarrow\\exists$ and cha... | Key idea: Negating quantified statements requires flipping each quantifier and negating the predicate while keeping variable order. Always sanity-check by translating to plain English. (Here the result is $\boxed{\exists x\in\mathbb{R}$.) |
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.