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).
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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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).
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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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