Explanation Given information: A,B and C be any sets. Concept used: A Δ B = ( A − B ) ∪ ( B − A ) . Calculation: The following are the Venn diagrams for A Δ C and B Δ C : The shaded portions denote the area covered by the symmetric difference. A Δ C = ( A − C ) ∪ ( C − A ) B Δ C = ( B − C ) ∪ ( C − B ) From the Venn diagram, it can be seen that if an element x ∈ A then it is present in either of the two regions: x ∈ A − C x ∈ A ∩ C Analyze both the above cases as follows: Case 1: x ∈ A − C The following two things are observed in this case using the above Venn diagram: x ∉ C Because x ∈ A − C x ∈ A Δ C Because the region where x belongs to is contained in the A Δ C region. Since it is given that A Δ C = B Δ C therefore x ∈ B Δ C . But x ∉ C , therefore, x must belong to B . Case2: x ∈ A ∩ C The following two things are observed in this case using the above Venn diagram: x ∈ C because x ∈ A ∩ C x ∉ A Δ C because the region where x belongs to is not contained in the A Δ C region . Since, it is given that A Δ C = B Δ C , therefore x ∉ ( B Δ C ) either . But x ∈ C , therefore, x must belong to B ∩ C because that is the only region that contains C But is itself not contained in ( B Δ C ) therefore x ∈ B . So, it is concluded that every element that belongs to A also belongs to B . Thus, A is a subset of B A ⊆ B

5th Edition

ISBN: 9780357097717

Author: EPP

Publisher: CENGAGE L

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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”.

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Chapter 6.3, Problem 51ES

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To prove:

If AΔC=BΔC, then A=B.

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Chapter 6.3, Problem 50ES

Chapter 6.3, Problem 52ES

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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 - 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: If A Δ C = B Δ C , then A = B .

Solution Summary: The author analyzes the Venn diagrams for ADelta C, , and andB, denote the area covered by the symmetric difference.

Concept explainers

To prove:

If AΔC=BΔC, then A=B.

Want to see the full answer?

Chapter 6 Solutions

To prove: If A Δ C = B Δ C , then A = B .

Solution Summary: The author analyzes the Venn diagrams for ADelta C, , and andB, denote the area covered by the symmetric difference.

Concept explainers

To prove: If AΔC=BΔC, then A=B.

Want to see the full answer?

Chapter 6 Solutions

To prove:

If AΔC=BΔC, then A=B.