1. [12 pts] The symmetric difference of sets A and B, denoted by AAB, is the...

  

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1. [12 pts] The symmetric difference of sets A and B, denoted by AAB, is the set containing those elements in A or B, but not in both A and B. For example, for A = {1, 2, 3} and B = {2, 3, 4, 5}, A A B = {1, 4, 5}. 1. Prove that (AA B) A B = A. 2. Is the following true: A A (BAC) = (AA B) A C? Prove your answer. 2. [8 pts] Prove that if A and B are two finite sets, then |AB| |AU B. Determine when this relationship is an equality. 3. [10 pts] An ordered pair (a, b) differs from a set of two elements {a, b} since the elements of a set are unordered. But we can represent an ordered pair in terms of sets. Introduce a set-based formulation of ordered pairs and prove that (a, b) = (c, d) under your set-based formulation if and only if a = c and b = d. 1. [12 pts] The symmetric difference of sets A and B, denoted by AAB, is the set containing those elements in A or B, but not in both A and B. For example, for A = {1, 2, 3} and B = {2, 3, 4, 5}, A A B = {1, 4, 5}. 1. Prove that (AA B) A B = A. 2. Is the following true: A A (BAC) = (AA B) A C? Prove your answer. 2. [8 pts] Prove that if A and B are two finite sets, then |AB| |AU B. Determine when this relationship is an equality. 3. [10 pts] An ordered pair (a, b) differs from a set of two elements {a, b} since the elements of a set are unordered. But we can represent an ordered pair in terms of sets. Introduce a set-based formulation of ordered pairs and prove that (a, b) = (c, d) under your set-based formulation if and only if a = c and b = d.

Answer rating: 100% (QA)

1 Prove that A B B A To prove this we need to show that for any element x x A B B if and only if x A Lets consider the cases Case 1 x A B B implies x A If x A B B it means that x is in either A B or B ... View the full answer