need help Part 2: Proof that (An C) - BC (AB) n (C - B)Consider the sentences in the following scrambled list.So by definition of set difference, x E A - B and x E C – B.By definition of set difference, x E A and xEC.Thus, by definition of intersection, x E A and x € C, and, in addition, x & B.Hence both x EA and x & B and also x E C, and x & B.By definition of intersection, x E (A – B) n (C – B).-By definition of intersection x EA n C and x & B.By definition of set difference x E A n C and x & B.To prove Part 2, select sentences from the list and put them in the correct order.1. Suppose x E (An C) - B.2. -Select---3. ---Select---4. --Select---5. --Select---6. --Select---7. Hence, (A n C) - BC (A - B) n (CB) by definition of subset.Conclusion:Since both subset relations have been proved, it follows by definition of set equality that (A - B) n (C − B) = (An C) - B. Use an element argument to prove the statement. (Assume that all sets are subsets of a universal set U.)Statement: For all sets A, B, and C,(A - B)n (CB) = (An C) – B.Proof:Suppose A, B, and C are any sets. [To show that (A − B) n (C – B) = (A n C) - B, we must show that (A - B) n (CB) C (An C) - B and that (A n C) - BC (A - B) n (C – B).]Part 1: Proof that (A - B)n (C - B) ≤ (An C) - BConsider the sentences in the following scrambled list.By definition of intersection, x E A - B and X E C - B.By definition of set difference, x E A - B and x E C - B.Therefore x E (An C) - B by the definition of set difference.By definition of set difference, x EA and x & B and x E C and x & B.By definition of intersection, x E A and x & B and X E C and x & B.Thus x E An C by definition of intersection, and, in addition, x & B.To prove Part 1, select sentences from the list and put them in the correct order.1. Suppose x € (A – B) n (C − B).2. Thus x EAN C by definition of intersection, and, in addition, x € B.3. --Select---4.---Select---5. -Select---6. Hence, (A - B) n (C - B) ≤ (An C) - B by definition of subset.Part 2: Proof that (A n C) - BC (A - B)n (C - B)Consider the sentences in the following scrambled list.So by definition of set difference, x E A - B and x E C - B.By definition of set difference, x E A and x E C.Thus, by definition of intersection, x EA and x E C, and, in addition, x & B.Hence both x E A and x & B and also x E C, and x & B.By definition of intersection, x E (A − B) n (C – B).By definition of intersection x EA n C and x & B.By definition of set difference x EA n C and x & B.

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Use an element argument to prove the statement. (Assume that all sets are subsets of a universal set U.) Statement: For all sets A, B, and C, (AB) n (CB) = (A nC) - B. Proof: Suppose A, B, and C are any sets. [To show that (AB) n (CB) = (An C) - B, we must show that (A - B) n (CB) C (An C) - B and that (An C) - BC (AB) n (C-B).] Part 1: Proof that (A - B)n (CB) C (An C) - B Consider the sentences in the following scrambled list. By definition of intersection, x EA - B and x EC - B. By definition of set difference, x EA - B and x EC - B. Therefore x E (An C) - B by the definition of set difference. By definition of set difference, x EA and x B and x E C and x € B. By definition of intersection, x EA and x B and x E C and x € B. Thus x E An C by definition of intersection, and, in addition, x # B. To prove Part 1, select sentences from the list and put them in the correct order. 1. Suppose x E (AB) n (CB). 2. Thus x € An C by definition of intersection, and, in addition, x # B. 3. ---Select--- 4. ---Select--- 5. ---Select--- 6. Hence, (AB) n (CB) C (An C) - B by definition of subset. Part 2: Proof that (An C) - BC (AB) n (CB) Consider the sentences in the following scrambled list. So by definition of set difference, x EA - B and x E C - B. By definition of set difference, x EA and x E C. Thus, by definition of intersection, x E A and x E C, and, in addition, x € B. Hence both x EA and x B and also x E C, and x # B. By definition of intersection, x E (A - B) n (CB). By definition of intersection x E An C and x € B. By definition of set difference x E An C and x € B.

8th Edition

ISBN:9781285463230

Author:Gilbert, Linda, Jimmie

Publisher:Gilbert, Linda, Jimmie

Chapter1: Fundamentals

Section1.1: Sets

Problem 1TFE: True or False Label each of the following statements as either true or false. Two sets are equal if...