   # 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).] ### Discrete Mathematics With Applicat...

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

#### Solutions

Chapter
Section ### Discrete Mathematics With Applicat...

5th Edition
EPP + 1 other
Publisher: Cengage Learning,
ISBN: 9781337694193
Chapter 6.2, Problem 6ES
Textbook Problem
1 views

## 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).

### Still sussing out bartleby?

Check out a sample textbook solution.

See a sample solution

#### The Solution to Your Study Problems

Bartleby provides explanations to thousands of textbook problems written by our experts, many with advanced degrees!

Get Started

Find more solutions based on key concepts
Simplify: 3064

Elementary Technical Mathematics

In Exercises 1520, simplify the expression. 15. 4(x2+y)3x2+y

Applied Calculus for the Managerial, Life, and Social Sciences: A Brief Approach

sec1(23)= a) 3 b) 3 c) 6 d) 6

Study Guide for Stewart's Single Variable Calculus: Early Transcendentals, 8th

In Exercises 1-6, solve for y. (x2)2+(y+1)2=9

Calculus: An Applied Approach (MindTap Course List)

Rationalize Numerator Rationalize the numerator. 92. 3+52

Precalculus: Mathematics for Calculus (Standalone Book)

Define the concept of internal validity and a threat to internal validity.

Research Methods for the Behavioral Sciences (MindTap Course List)

5.5 SKILL BUILDING EXERCISES Using the Quadratic formula In Exercises S-1 through S-6, use the quadratic formul...

Functions and Change: A Modeling Approach to College Algebra (MindTap Course List)

For 0

Study Guide for Stewart's Multivariable Calculus, 8th

Solve the equations in Exercises 126. x4x+1xx1=0

Finite Mathematics and Applied Calculus (MindTap Course List) 