Homework Help is Here – Start Your Trial Now!
Math
Advanced Math
to prove the Statement. (Assume that Statement: For all sets A, B, and C, (A - B) n (C - B) = (A N 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 (C – B) C (A N C) – Band that (A n C) - BE (A - B) n (C - B).] Part 1: Proof that (A - B) N (C – B) C (A N C) - B Consider the sentences in the following scrambled list. Thus x E ANC by definition of intersection, and, in addition, x € B. Therefore x E (AN C) - B by the definition of set difference. By definition of set difference, x E A - Band x EC - B. By definition of intersection, x E A and x EB and x E C and x € B. By definition of intersection, x E A - B and x EC - B. By definition of set difference, x E A and x € B and x E C and 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. --Select--. 3. ---Select-. 4. ---Select--- 5. ---Select-. 6. Hence, (A - B) n (C - B) S (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. By definition of intersection, x € (A - B) n (C - B). By definition of intersection x E AnC and x € B. By definition of set difference, x E A and x E C. By definition of set difference x E ANC and x ¢ B. Hence both x E A and x € B and also x E C, and x € B. So by definition of set difference, x E A - Band x E C - B. Thus, by definition of intersection, x E A and x € C, and, in addition, 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.-. -Select--- 6. --Select--- 7. Hence, (A N C) - BE (A - B) n (C - B) by definition of subset. 5.

to prove the Statement. (Assume that Statement: For all sets A, B, and C, (A - B) n (C - B) = (A N 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 (C – B) C (A N C) – Band that (A n C) - BE (A - B) n (C - B).] Part 1: Proof that (A - B) N (C – B) C (A N C) - B Consider the sentences in the following scrambled list. Thus x E ANC by definition of intersection, and, in addition, x € B. Therefore x E (AN C) - B by the definition of set difference. By definition of set difference, x E A - Band x EC - B. By definition of intersection, x E A and x EB and x E C and x € B. By definition of intersection, x E A - B and x EC - B. By definition of set difference, x E A and x € B and x E C and 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. --Select--. 3. ---Select-. 4. ---Select--- 5. ---Select-. 6. Hence, (A - B) n (C - B) S (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. By definition of intersection, x € (A - B) n (C - B). By definition of intersection x E AnC and x € B. By definition of set difference, x E A and x E C. By definition of set difference x E ANC and x ¢ B. Hence both x E A and x € B and also x E C, and x € B. So by definition of set difference, x E A - Band x E C - B. Thus, by definition of intersection, x E A and x € C, and, in addition, 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.-. -Select--- 6. --Select--- 7. Hence, (A N C) - BE (A - B) n (C - B) by definition of subset. 5.

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 1TFE: True or False Label each of the following statements as either true or false. Two sets are equal if...
See similar textbooks
icon
Related questions
Question
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 (C – B) = (A N 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 (C – B) C (A N C) – B and that (A n C) – BC (A – B) N (C – B).] Part 1: Proof that (A - B) N (C – B) C (A n C) – B Consider the sentences in the following scrambled list. Thus x E AN C by definition of intersection, and, in addition, x € B. Therefore x € (A N C) – B by the definition of set difference. By definition of set difference, x E A - B and x E C - B. By definition of intersection, x E A and x €B and x E C and x € B. By definition of intersection, x E A - B and x E C - B. By definition of set difference, x E A and x É B and x E C and x € B. To prove Part 1, select sentences from the list and put them in the correct order. 1. Suppose x E (A – B) N (C - B). 2. --Select--- 3. --Select-- 4. --Select--- 5. --Select-- 6. Hence, (A – B) N (C – B) S (A N 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. By definition of intersection, x € (A – B) (C – B). By definition of intersection x € A NC and x ¢ B. By definition of set difference, x E A and x E C. By definition of set difference x E ANC and x € B. Hence both xE A and x € B and also x E C, and x ¢ B. So by definition of set difference, x E A – B and x E C - B. Thus, by definition of intersection, x E A and x € C, and, in addition, x € B. To prove Part 2, select sentences from the list and put them in the correct order. 1. Suppose x E (A N C) – B. 2. --Select--- 3. -Select--- 4. --Select--- 5. ---Select- 6. ---Select--- 7. Hence, (A n c) - BC (A – B) N (C – B) by definition of subset.
Transcribed Image Text: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 (C – B) = (A N 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 (C – B) C (A N C) – B and that (A n C) – BC (A – B) N (C – B).] Part 1: Proof that (A - B) N (C – B) C (A n C) – B Consider the sentences in the following scrambled list. Thus x E AN C by definition of intersection, and, in addition, x € B. Therefore x € (A N C) – B by the definition of set difference. By definition of set difference, x E A - B and x E C - B. By definition of intersection, x E A and x €B and x E C and x € B. By definition of intersection, x E A - B and x E C - B. By definition of set difference, x E A and x É B and x E C and x € B. To prove Part 1, select sentences from the list and put them in the correct order. 1. Suppose x E (A – B) N (C - B). 2. --Select--- 3. --Select-- 4. --Select--- 5. --Select-- 6. Hence, (A – B) N (C – B) S (A N 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. By definition of intersection, x € (A – B) (C – B). By definition of intersection x € A NC and x ¢ B. By definition of set difference, x E A and x E C. By definition of set difference x E ANC and x € B. Hence both xE A and x € B and also x E C, and x ¢ B. So by definition of set difference, x E A – B and x E C - B. Thus, by definition of intersection, x E A and x € C, and, in addition, x € B. To prove Part 2, select sentences from the list and put them in the correct order. 1. Suppose x E (A N C) – B. 2. --Select--- 3. -Select--- 4. --Select--- 5. ---Select- 6. ---Select--- 7. Hence, (A n c) - BC (A – B) N (C – B) by definition of subset.
Expert Solution
trending now

Trending now

This is a popular solution!

steps

Step by step

Solved in 3 steps with 3 images

SEE SOLUTIONCheck out a sample Q&A here
Blurred answer