Use an element argument to prove the following statement.. Proof: For all sets A, B, and C, An (B-C) ≤ (ANB) - (ANC). 1. Suppose A, B, and C are any sets. [To show that An (B-C) ≤ (ANB) - (An C), we must show that for every element x, if x € --Select--- 2. Suppose x € ---Select-- v [We must show that x E ---Select--- 3. By definition of intersection, x EA ---Select--xEB-C. 4. By the set difference law, xEB-Select--- ✓x € C. [By step 3, we also know that x € ---Select--- .] 5. Thus, x E A and both x E B and x ---Select-- 6. By step 5, x EA and x E B, and thus x E---Select--- 7. Also by step 5, x EA and x € C, and thus x ---Select--- 9. Hence, by definition of subset, ✓by definition of ---Select--- [Why? If x E An C, then, by definition of ---Select--- 8. Thus x EAN B and x @ An C, and so, by the set difference law, x E---Select--- [This shows that every element in ---Select--- is in ---Select--- -Select-- by definition of ---Select--- ✓, x would be in C, which it is not.] ---Select--- V. V. ✓, then x E-Select--
Use an element argument to prove the following statement.. Proof: For all sets A, B, and C, An (B-C) ≤ (ANB) - (ANC). 1. Suppose A, B, and C are any sets. [To show that An (B-C) ≤ (ANB) - (An C), we must show that for every element x, if x € --Select--- 2. Suppose x € ---Select-- v [We must show that x E ---Select--- 3. By definition of intersection, x EA ---Select--xEB-C. 4. By the set difference law, xEB-Select--- ✓x € C. [By step 3, we also know that x € ---Select--- .] 5. Thus, x E A and both x E B and x ---Select-- 6. By step 5, x EA and x E B, and thus x E---Select--- 7. Also by step 5, x EA and x € C, and thus x ---Select--- 9. Hence, by definition of subset, ✓by definition of ---Select--- [Why? If x E An C, then, by definition of ---Select--- 8. Thus x EAN B and x @ An C, and so, by the set difference law, x E---Select--- [This shows that every element in ---Select--- is in ---Select--- -Select-- by definition of ---Select--- ✓, x would be in C, which it is not.] ---Select--- V. V. ✓, then x E-Select--
Calculus For The Life Sciences
2nd Edition
ISBN:9780321964038
Author:GREENWELL, Raymond N., RITCHEY, Nathan P., Lial, Margaret L.
Publisher:GREENWELL, Raymond N., RITCHEY, Nathan P., Lial, Margaret L.
Chapter12: Probability
Section12.CR: Chapter 12 Review
Problem 3CR
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Question
![Use an element argument to prove the following statement.
Proof:
For all sets A, B, and C, An (B-C) ≤ (ANB) - (ANC).
1. Suppose A, B, and C are any sets.
[To show that An (B-C) ≤ (ANB) - (An C), we must show that for every element x, if x E ---Select---
2. Suppose xE---Select---
[We must show that x E---Select---
✓.1
3. By definition of intersection, x E A ---Select--- XEB-C.
4. By the set difference law, x E B ---Select--- x # C.
[By step 3, we also know that x E|---Select--- .]
5. Thus, x E A and both x E B and x # ---Select---
6. By step 5, x EA and x E B, and thus x E---Select---
7. Also by step 5, x EA and x # C, and thus x # ---Select---
[Why? If x EAN C, then, by definition of ---Select---
8. Thus x E An Band x @ An C, and so, by the set difference law, x E ---Select---
[This shows that every element in ---Select---
is in --Select---
9. Hence, by definition of subset, ---Select---
by definition of ---Select---
V ---Select---
by definition of ---Select---
x would be in C, which it is not.]
.]
then x E---Select---
.]](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fa0e38307-1ade-44bc-b712-aaeda4c58098%2F329bb073-d745-4167-97c1-f0ce488a6ba8%2Fw26wirh_processed.jpeg&w=3840&q=75)
Transcribed Image Text:Use an element argument to prove the following statement.
Proof:
For all sets A, B, and C, An (B-C) ≤ (ANB) - (ANC).
1. Suppose A, B, and C are any sets.
[To show that An (B-C) ≤ (ANB) - (An C), we must show that for every element x, if x E ---Select---
2. Suppose xE---Select---
[We must show that x E---Select---
✓.1
3. By definition of intersection, x E A ---Select--- XEB-C.
4. By the set difference law, x E B ---Select--- x # C.
[By step 3, we also know that x E|---Select--- .]
5. Thus, x E A and both x E B and x # ---Select---
6. By step 5, x EA and x E B, and thus x E---Select---
7. Also by step 5, x EA and x # C, and thus x # ---Select---
[Why? If x EAN C, then, by definition of ---Select---
8. Thus x E An Band x @ An C, and so, by the set difference law, x E ---Select---
[This shows that every element in ---Select---
is in --Select---
9. Hence, by definition of subset, ---Select---
by definition of ---Select---
V ---Select---
by definition of ---Select---
x would be in C, which it is not.]
.]
then x E---Select---
.]
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