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"from": "human",
"value": "**Problem:**\n\nLet \\( X = [0, 1] \\cup [2, 3] \\) be a subset of the real numbers \\( \\mathbb{R} \\) with the standard topology. \n\n1. Prove that \\( X \\) is a disconnected space.\n2. Identify the connected components of \\( X \\).\n3. Determine whether \\( X \\) is compact... |
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"from": "human",
"value": "Sure! Here鈥檚 a problem that focuses on counting permutations of a set with repeated elements:\n\n---\n\n**Problem:**\n\nA box contains 5 different colored balls: 2 red, 2 blue, and 1 green. How many distinct ways can you arrange all the balls in a row?\n\n**Instructions:**\n1. I... |
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"from": "human",
"value": "Consider a box containing 6 balls: 3 red, 2 blue, and 1 green. \n\n**Question:** In how many distinct ways can you arrange all the balls in a single row?\n\n**Instructions:** Show your calculations and explain each step in your solution."
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"from": "gpt",
"value":... |
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"from": "human",
"value": "Sure! Here鈥檚 a combinatorics problem involving constraints for a student to solve:\n\n**Problem:**\n\nA committee of 6 members is to be formed from a group of 10 candidates: 4 men and 6 women. However, the committee must include at least 2 men and at least 3 women.\n\n1. How man... |
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"from": "human",
"value": "Sure! Here's a problem related to counting permutations of a set with repeated elements:\n\n---\n\n**Problem:**\nA box contains 8 colored balls: 3 red, 2 blue, and 3 green. \n\n1. In how many different ways can the balls be arranged in a single row?\n2. If two of the red balls a... |
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"from": "human",
"value": "Sure! Here's a combinatorics problem with constraints for a student to solve:\n\n---\n\n**Problem:**\n\nA school club has 8 members: Alice, Bob, Charlie, David, Eve, Fiona, George, and Hannah. The club needs to form a committee of 5 members to organize a school event.\n\nHowever... |
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"from": "human",
"value": "**Problem:**\n\nLet \\( X = [0, 1] \\) be the closed interval in the standard topology on \\( \\mathbb{R} \\). \n\n1. Prove that \\( X \\) is compact.\n2. Consider the set \\( Y = [0, 1) \\) with the subspace topology induced from \\( \\mathbb{R} \\). Is \\( Y \\) compact? Justi... |
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{
"from": "human",
"value": "**Problem: Compactness and Connectedness in Topological Spaces**\n\nLet \\( X = [0, 1] \\cup [2, 3] \\) be equipped with the standard topology inherited from \\( \\mathbb{R} \\). \n\n1. Prove whether \\( X \\) is a compact space.\n2. Determine if \\( X \\) is connected. If not, ... |
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"from": "human",
"value": "### Problem:\n\nLet \\( X \\) be a topological space that is path-connected and locally contractible. Define the fundamental group \\( \\pi_1(X, x_0) \\) of \\( X \\) based at a point \\( x_0 \\in X \\).\n\n1. Prove that for any two paths \\( \\gamma_1, \\gamma_2: [0,1] \\to X \... |
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{
"from": "human",
"value": "**Problem:**\n\nLet \\( X \\) be a topological space and consider the following scenarios:\n\n1. Let \\( A \\) and \\( B \\) be two subspaces of \\( X \\). Assume that \\( A \\) is a deformation retract of \\( B \\). Show that the inclusion map \\( i: A \\hookrightarrow B \\) in... |
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"from": "human",
"value": "**Problem:**\n\nA teacher has a set of 10 different books and wants to create a reading list for a semester. However, the reading list must meet the following constraints:\n\n1. The reading list must consist of exactly 5 books.\n2. At least 2 of the selected books must be from t... |
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{
"from": "human",
"value": "**Problem:**\n\nLet \\( X \\) be a topological space and \\( A \\subseteq X \\) a subspace. Consider the inclusion map \\( i: A \\hookrightarrow X \\).\n\n1. Define the first homotopy group \\( \\pi_1(X, x_0) \\) and the first homotopy group of the subspace \\( A \\), \\( \\pi_1... |
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