Explanation Given information: Let A,B a n d C be any set Concept used: ∪ : U n i o n of sets ∩ : I n t e r sec t i o n of sets ⊆ : subset of set Calculation: Consider the statement, If A ∩ C ⊆ B ∩ C and A ∪ C ⊆ B ∪ C . The objective is to verify whether the statement is true or not. Recall, the definition for union, and intersection, Suppose A and B are two subsets of a universal set ∪ . The union of A and B is denoted by A ∪ B , and is defined by, A ∪ B = { x ∈ U | x ∈ A , or, x ∈ B } The intersection of A and B is denoted by A ∩ B , and is defined by A ∩ B = { x ∈ U | x ∈ A , and, x ∈ B } . Let A , B and C is any sets. Suppose that if x ∈ A when x ∈ C or x ∉ C . Case 1. For x ∈ A and x ∈ C

5th Edition

ISBN: 9780357035283

Author: EPP

Publisher: Cengage

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Set Theory

Every field in the present world involves a set. The theory of sets was developed by German mathematician Georg Cantor while working on the “problems of trigonometric series”.

Question

Chapter 6.3, Problem 14ES

To determine

To Prove:

For each prove each statement that is true and find a counterexample for each statement that is false. Assume all sets are subsets of a universal set U.

For all sets A,B and C

if A∩C⊆B∩C and A∪C⊆B∪C then A⊆B.

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Chapter 6.3, Problem 13ES

Chapter 6.3, Problem 15ES

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Chapter 6 Solutions

Discrete Mathematics With Applications

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 - Prob. 3TY 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 - Prob. 10TY

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To Prove: For each prove each statement that is true and find a counterexample for each statement that is false. Assume all sets are subsets of a universal set U . For all sets A , B and C if A ∩ C ⊆ B ∩ C and A ∪ C ⊆ B ∪ C then A ⊆ B .

Solution Summary: The author explains that the objective is to verify whether the statement is true or not.

Concept explainers

To Prove:

For each prove each statement that is true and find a counterexample for each statement that is false. Assume all sets are subsets of a universal set U.

For all sets A,B and C

if A∩C⊆B∩C and A∪C⊆B∪C then A⊆B.

Want to see the full answer?

Chapter 6 Solutions

To Prove: For each prove each statement that is true and find a counterexample for each statement that is false. Assume all sets are subsets of a universal set U . For all sets A , B and C if A ∩ C ⊆ B ∩ C and A ∪ C ⊆ B ∪ C then A ⊆ B .

Solution Summary: The author explains that the objective is to verify whether the statement is true or not.

Concept explainers

To Prove: For each prove each statement that is true and find a counterexample for each statement that is false. Assume all sets are subsets of a universal set U. For all sets A,B and C if A∩C⊆B∩C and A∪C⊆B∪C then A⊆B.

Want to see the full answer?

Chapter 6 Solutions

To Prove:

For each prove each statement that is true and find a counterexample for each statement that is false. Assume all sets are subsets of a universal set U.

For all sets A,B and C

if A∩C⊆B∩C and A∪C⊆B∪C then A⊆B.