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Discrete Mathematics With Applications5th Edition
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Use an element argument to prove each statement in 7—22. Assume that all sets are subsets of a universal set ∪ . 10. For all sets A , B , and C , ( A ∪ B ) ∩ C ⊆ A ∪ ( B ∩ C )

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Discrete Mathematics With Applicat...

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

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Discrete Mathematics With Applicat...

5th Edition
EPP + 1 other
Publisher: Cengage Learning,
ISBN: 9781337694193
Chapter 6.2, Problem 10ES
Textbook Problem
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Use an element argument to prove each statement in 7—22. Assume that all sets are subsets of a universal set ∪ .

10. For all sets A, B, and C, ( A ∪ B ) ∩ C ⊆ A ∪ ( B ∩ C )

To determine

To prove:

For all sets A ,B and C, (A−B)∩(C−B)=(A∩C)−B

Explanation of Solution

Given information:

A And B be two sets.

Concept used:

∪:Union of sets∩:Intersection of sets

Calculation:

Let A ,B and C are any three sets.

Prove that (A−B)∩(C−B)=(A∩C)−B.

First prove that (A−B)∩(C−B)⊆(A∩C)−B.

Suppose there is an element x in (A−B)∩(C−B), by definition of union x∈A−B and x∈C−B.

When x∈A−B then by set difference x∈A and x∉B.

When x∈C−B then by set difference x∈C and x∉B.

Since x∈A and x∈C, by definition of intersection it follows that x∈A∩C.

Thus, x∈A∩C and x∉B, the definition of set difference implies that x∈(A∩C)−B.

Hence x∈(A−B)∩(C−B)⇒x∈(A∩C)−B.

Therefore, (A−B)∩(C−B)⊆(A∩C)−B

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Ch. 6.1 - The notation is read”______” and means that___Ch. 6.1 - To use an element argument for proving that a set...Ch. 6.1 - To disprove that a set X is a subset of set Y, you...Ch. 6.1 - An element x is in AB if , and only if,_______Ch. 6.1 - An element x in AB if, and only if,______Ch. 6.1 - An element x is in B-A if, and only if,______Ch. 6.1 - An elements x is in Acif, and only if.______Ch. 6.1 - The empty set is a set with ______Ch. 6.1 - The power set of a set A is _____Ch. 6.1 - Sets A and B are disjoint if, and only...
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 ={} ? Why ? c....Ch. 6.1 - Let Ai={i,i2} for each integer i=1,2,3,4. a....Ch. 6.1 - Let Bi={xR0xi} for each integer i=1,2,3,4. a....Ch. 6.1 - Let Ci={i,i} for each nonnegative integer i.Ch. 6.1 - Let Di={xR-ixi}=[i,i] for each nonnegative integer...Ch. 6.1 - Let Vi={xR1ix1i}=[1i,1i] for each positive integer...Ch. 6.1 - Let Wi={xRxi}=(i,) for each nonnegative integer i....Ch. 6.1 - Let Ri={xR1x1+1i}=[1,1+1i]foreachpositiveintegeri....Ch. 6.1 - Let Si={xR1x1+1i}=(1,1+1i) for each positive...Ch. 6.1 - a. Is {{a, d, e}, {b, c}, {d, f }} a partition of...Ch. 6.1 - Let E be the set of all even integers and O the...Ch. 6.1 - Let R be the set of all real number. Is a...Ch. 6.1 - Let Z be the set of all integers and let...Ch. 6.1 - Suppose A={1,2} and B={2,3} . Find each of the...Ch. 6.1 - Suppose A={1} and B={u,v} . Find P(AB) . Suppose...Ch. 6.1 - Find P() FindP(p()). Find p(p(p())) .Ch. 6.1 - Let A1={1},A2={u,v},andA3={m,n}. Find each of the...Ch. 6.1 - Let...Ch. 6.1 - Trace the action of Algorithm 6,1,1 on the...Ch. 6.1 - Trace the action of Algorithm 6,1,1 on the...Ch. 6.1 - Write an algorithm to determine whether a given...Ch. 6.2 - To prove that a set X is a subset of a set you...Ch. 6.2 - To prove that a set X is a subset of a set AB, you...Ch. 6.2 - To prove that a set ABis a subset of a set X, you...Ch. 6.2 - To prove that a set AB is a subset of a set X, you...Ch. 6.2 - To prove that a set X equals a set Y, you prove...Ch. 6.2 - To prove that a set X does not equal a set Y, you...Ch. 6.2 - To say that an element is in A(BC) means that it...Ch. 6.2 - The following are two proofs that for all sets A...Ch. 6.2 - In 3 and 4, supply explanations of the steps in...Ch. 6.2 - Theorem: For all sets A and B, if AB , then ABB.Ch. 6.2 - Prove that for all set A and B, (BA)=BAe .Ch. 6.2 - Let and stand for the words “intersection” and...Ch. 6.2 - Use an element argument to prove each statement in...Ch. 6.2 - Use an elements argument to prove each statement...Ch. 6.2 - Use an element argument to prove each statement in...Ch. 6.2 - Use an element argument to prove each statement in...Ch. 6.2 - Use an element argument to prove each statement in...Ch. 6.2 - Use an element argument to prove each statement in...Ch. 6.2 - Use an element argument to prove each statement in...Ch. 6.2 - Use an element argument to prove each statement in...Ch. 6.2 - Use an element argument to prove each statement in...Ch. 6.2 - Use an element argument to prove each statement in...Ch. 6.2 - Use an element argument to prove each statement in...Ch. 6.2 - Use an element argument to prove each statement in...Ch. 6.2 - Use an element argument to prove each statement in...Ch. 6.2 - Use an element argument to prove each statement in...Ch. 6.2 - Use an element argument to prove each statement in...Ch. 6.2 - Use an element argument to prove each statement in...Ch. 6.2 - Find the mistake in the following : proof” that...Ch. 6.2 - Find the mistake in all the following “proof.”...Ch. 6.2 - Find the mistake in the following “proof” that for...Ch. 6.2 - Consider the Venn diagram below. Illustrate one of...Ch. 6.2 - Fill in the blanks in the following proof that for...Ch. 6.2 - Use the element method for proving a set equals...Ch. 6.2 - Use the element method for proving a set equals...Ch. 6.2 - Use an element argument to prove each statement in...Ch. 6.2 - Use an element argument to prove each statement in...Ch. 6.2 - Use an element argument to prove each statement in...Ch. 6.2 - Use an element argument to prove each statement in...Ch. 6.2 - Use an element argument to prove each statement in...Ch. 6.2 - Use an element argument to prove each statement in...Ch. 6.2 - Use an element argument to prove each statement in...Ch. 6.2 - Use an element argument to prove each statement in...Ch. 6.2 - Use an element argument to prove each statement in...Ch. 6.2 - Prove each statement is 39-44. For all sets A and...Ch. 6.2 - Prove each statement in 39-44. For every positive...Ch. 6.2 - Prove each statement in 39-44. For every positive...Ch. 6.2 - Prove each statement in 39-44. 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