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Let ∩ and ∪ stand for the words “intersection” and “union,” respectively. Fill in the blanks in the following proof that for all sets A, B, and C, A ∩ ( B ∪ C ) = ( A ∩ B ) ∪ ( A ∩ C ) . Proof: Suppose A , B and C are any sets. (1) Proof that A ∩ ( B ∩ C ) ⊆ ( A ∩ B ) ∩ ( A ∩ C ) : Let x ∈ A ∩ ( B ∪ C ) . [We must show that x ∈ (a) ]. By definition of ∩ , x ∈ (b) and x ∈ B ∪ C . Thus x ∈ A and, by definition of ∪ , x ∈ B or (c) . Case 1 ( x ∈ A and x ∈ B ): In this case, x ∈ A ∩ B by definition of ∪ . Case 2 ( x ∈ A and x ∈ C ): In this case, x ∈ A ∩ C by definition of ∪ . By cases 1 and 2, x ∈ A ∩ B or x ∈ A ∩ C , and so, by definition of ∪ . (d) . [So A ∩ ( B ∪ C ) ⊆ ( A ∩ B ) ∪ ( A ∩ C ) by definition of subset.] (2) Proof that ( A ∩ B ) ∪ ( A ∩ C ) ⊆ A ∩ ( B ∪ C ) : Let x ∈ ( A ∩ B ) ∪ ( A ∩ C ) . [We must show that x ∈ A ∩ ( B ∪ C ) .] By definition of ∪ , x ∈ A ∩ B (a) x ∈ A ∩ C . Case 1 ( x ∈ A ∩ B ) : In this case, by definition of ∪ , x ∈ A and x ∈ B . Since x ∈ B , then x ∈ B ∪ C by definition of ∪ . Case 2 ( x ∈ A ∩ C ) : In this case, by definition of ∪ , x ∈ A (b) x ∈ C . Since x ∈ C , then x ∈ B ∪ C by definition of ∪ . In both cases x ∈ A and x ∈ B ∪ C , and so, by definition of ∪ , (c) . [So ( A ∩ B ) ∪ ( A ∩ C ) ⊆ A ∩ ( B ∩ C ) by definition of (d) .] (3) Conclusion: [Since both subset relations have been proved, it follows, by definition of set equality, that (a).]

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Discrete Mathematics With Applicat...

5th Edition
EPP + 1 other
Publisher: Cengage Learning,
ISBN: 9781337694193

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BuyFindarrow_forward

Discrete Mathematics With Applicat...

5th Edition
EPP + 1 other
Publisher: Cengage Learning,
ISBN: 9781337694193
Chapter 6.2, Problem 6ES
Textbook Problem
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Let and stand for the words “intersection” and “union,” respectively. Fill in the blanks in the following proof that for all sets A, B, and C, A ( B C ) = ( A B ) ( A C ) .

Proof: Suppose A, B and C are any sets.

(1) Proof that A ( B C ) ( A B ) ( A C ) :

Let x A ( B C ) . [We must show that x (a) ].

By definition of , x (b) and x B C .

Thus x A and, by definition of , x B or (c) .

Case 1 ( x A and x B ): In this case, x A B by definition of .

Case 2 ( x A and x C ): In this case, x A C by definition of .

By cases 1 and 2, x A B or x A C , and so, by definition of . (d) .
[So A ( B C ) ( A B ) ( A C ) by definition of subset.]
(2) Proof that
( A B ) ( A C ) A ( B C ) :

Let x ( A B ) ( A C ) . [We must show that x A ( B C ) .]
By definition of , x A B (a) x A C .

Case 1 ( x A B ) : In this case, by definition of , x A and x B .

Since x B , then x B C by definition of .

Case 2 ( x A C ) : In this case, by definition of , x A (b) x C .

Since x C , then x B C by definition of .

In both cases x A and x B C , and so, by definition of , (c) .

[So ( A B ) ( A C ) A ( B C ) by definition of (d) .]
(3) Conclusion: [Since both subset relations have been proved, it follows, by definition of set equality, that (a).]

To determine

(a)

To fill:

The given blanks in the proof that forallthesetsA,BandC ,

A(BC)=(AB)(AC).

Explanation of Solution

Given:

Let xA(BC)

    Now we must show that ________(a)________

    By definition of , x __(b)___ and xBC

    Thus xA, and by definition of , x

    B or x __(c)___Case 1 (xAandxB): In this case xAB by definition of .

    Case 2 (xAandxC): In this case xAC by definition of .

    By cases 1 and 2, xAB or xAC, and so, by definition of ,______(d)________.

    [SoA(BC)(AB)(AC)bydefinitionofsubset]

Concept used:

    If an element xAB, then xAandxB

    If an element xAB, then xAorxB

    AB, if and only if every element of A is also an element of B.

    And A=B, if AB and BA .

Proof:

    Let xA(BC)

    Now we must show that x(AB)(AC)

    [ABx,ifxAthenxB]

    By definition of , x

    A and xBC

    [If an element,

To determine

(b)

To fill:

The given blanks in the proof that forallthesetsA,BandC ,

A(BC)=(AB)(AC).

To determine

(c)

To fill:

The given blanks in the proof that forallthesetsA,BandC ,

A(BC)=(AB)(AC).

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Chapter 6 Solutions

Discrete Mathematics With Applications
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Ch. 6.1 - A collection of nonempty set is a partition of a...Ch. 6.1 - In each of (a)-(f), answer the following question:...Ch. 6.1 - Complete the proof from Example 6.1.3: Prove that...Ch. 6.1 - Let sets R, S, and T be defined as follows:...Ch. 6.1 - Let A={nZn=5rforsomeintegerr} and...Ch. 6.1 - Let C={nZn=6r5forsomeintegerr} and...Ch. 6.1 - Let...Ch. 6.1 - ...Ch. 6.1 - Write in words how to end to read each of the...Ch. 6.1 - Complete the following sentences without using the...Ch. 6.1 - ...Ch. 6.1 - Let the universal set be R, the set of all real...Ch. 6.1 - Let the universal set be R, the set of all real...Ch. 6.1 - Let S be the set of all strings of 0’s and 1’s of...Ch. 6.1 - In each of the following, draw a Venn diagram for...Ch. 6.1 - In each of the following, draw a Venn diagram for...Ch. 6.1 - Let A={a,b,c},B={b,c,d} , and C={b,c,e} a. Find...Ch. 6.1 - Consider the following Venn diagram. For each of...Ch. 6.1 - a. Is the number 0 in ? Why? b. Is ={} ? 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