Sets

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Step 1: Find integers whose square is less than 20. Step 2: 0^2 = 0, 1^2 = 1, 2^2 = 4, 3^2 = 9, and 4^2 = 16 are all less than 20. Step 3: 5^2 = 25 is not less than 20, so stop there. Step 4: Include …

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Step 1: Notice that every element is an even natural number. Step 2: Each term can be written as 2n. Step 3: Since the set is finite, restrict n to the correct values. Answer: {x : x = 2n, n is a natu…

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Step 1: Check each item exactly as written. Step 2: 1 is an element of A, so 1 ∈ A is correct. Step 3: {1} is not listed as an element, so {1} ∈ A is incorrect. Step 4: {3, 4} is listed as one element…

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Step 1: Equal sets have exactly the same elements. Step 2: Equivalent sets have the same number of elements. Step 3: Example of equal sets: {1, 2, 3} and {3, 2, 1} are equal. Step 4: Example of equiva…

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Step 1: Check whether every element of A is in B. Step 2: 1, 2, and 3 are all in B, so A ⊆ B. Step 3: A is not equal to B because 4 is in B but not in A. Step 4: Since A ⊆ B and A ≠ B, A is a proper s…

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Step 1: Take any set A. Step 2: To prove φ ⊆ A, every element of φ must belong to A. Step 3: But φ has no elements. Step 4: So there is no element in φ that fails to belong to A. Answer: φ ⊆ A for eve…

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Step 1: List all subsets of A. Step 2: The subsets are φ, {a}, {b}, and {a, b}. Step 3: Collect them into one set. Answer: P(A) = {φ, {a}, {b}, {a, b}} Quick check: Every element of P(A) is a set.

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Step 1: Use the formula for number of subsets: 2^n. Step 2: Here n = 3, so number of subsets = 2^3 = 8. Step 3: Number of proper subsets = 2^n - 1. Step 4: So proper subsets = 8 - 1 = 7. Answer: 8 sub…