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Prove the following statement. Assume that all sets are subsets of a universal set U. For all sets A and B, P(AN B) = P(A) n P(8).

Prove the following statement. Assume that all sets are subsets of a universal set U. For all sets A and B, P(AN B) = P(A) n P(8).

Elements Of Modern Algebra
Elements Of Modern Algebra
8th Edition
ISBN:9781285463230
Author:Gilbert, Linda, Jimmie
Publisher:Gilbert, Linda, Jimmie
Chapter1: Fundamentals
Section1.1: Sets
Problem 6E: 6. Determine whether each of the following is either , , , or , where is an arbitrary subset of the...
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Part 2: Proof that P(A) n P(B) s P(AN B). To prove part 2, select options from the list and put them in the correct order. Statement Explanation Suppose A and B are any sets. 1. Let X be any element in ---Select-- -Select- 2. --Select-- for every x in X, xE A and xe B for every x in X, xE ANB Then Xe P(A) and Xe P(B) P(ANB) Then XANB P(A) N P(B) Hence, XA and XB ---Select- 3. -Select-- ---Select- 4. So, -Select--- ---Select- 5. Thus, Select-- -Select--- 6. ---Select-- ---Select- 7. Therefore, Xe Select-- -Select--- 8. Since X could be any element in --Select- | it follows that every element in -Select--- is in --Select--- 9. Therefore, P(A) n P(8) S P(AN B). Need Help? Read It Submit Answer
Transcribed Image Text:Part 2: Proof that P(A) n P(B) s P(AN B). To prove part 2, select options from the list and put them in the correct order. Statement Explanation Suppose A and B are any sets. 1. Let X be any element in ---Select-- -Select- 2. --Select-- for every x in X, xE A and xe B for every x in X, xE ANB Then Xe P(A) and Xe P(B) P(ANB) Then XANB P(A) N P(B) Hence, XA and XB ---Select- 3. -Select-- ---Select- 4. So, -Select--- ---Select- 5. Thus, Select-- -Select--- 6. ---Select-- ---Select- 7. Therefore, Xe Select-- -Select--- 8. Since X could be any element in --Select- | it follows that every element in -Select--- is in --Select--- 9. Therefore, P(A) n P(8) S P(AN B). Need Help? Read It Submit Answer
Prove the following statement. Assume that all sets are subsets of a universal set U. For all sets A and B, P(AN B) = P(A) N P(B). Proof: Consider the options in the following scrambled list. for every x in X, x E A and xeB for every x in X, x E ANB Then XE P(A) and Xe P(8) P(AN B) Then XCANB P(A) n P(B) Hence, XEA and XEB by definition of intersection by definition of power set by definition of subset Part 1: Proof that P(AN B) S P(A) N P(B). To prove part 1, select options from the list and put them in the correct order. Statement Explanation Suppose A and B are any sets. 1. Let X be any element in -Select- 2. ---Select-- ---Select- 3. So, -Select--- ---Select- 4. Thus, -Select- ---Select- 5. --Select-- -Select- 6. ---Select-- ---Select-- 7. Therefore, X E --Select-- ---Select- 8. Since X could be any element in --Select-- v it follows that every element in -Select--- v is in --Select--
Transcribed Image Text:Prove the following statement. Assume that all sets are subsets of a universal set U. For all sets A and B, P(AN B) = P(A) N P(B). Proof: Consider the options in the following scrambled list. for every x in X, x E A and xeB for every x in X, x E ANB Then XE P(A) and Xe P(8) P(AN B) Then XCANB P(A) n P(B) Hence, XEA and XEB by definition of intersection by definition of power set by definition of subset Part 1: Proof that P(AN B) S P(A) N P(B). To prove part 1, select options from the list and put them in the correct order. Statement Explanation Suppose A and B are any sets. 1. Let X be any element in -Select- 2. ---Select-- ---Select- 3. So, -Select--- ---Select- 4. Thus, -Select- ---Select- 5. --Select-- -Select- 6. ---Select-- ---Select-- 7. Therefore, X E --Select-- ---Select- 8. Since X could be any element in --Select-- v it follows that every element in -Select--- v is in --Select--
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