   Chapter 6.3, Problem 52ES ### Discrete Mathematics With Applicat...

5th Edition
EPP + 1 other
ISBN: 9781337694193

#### Solutions

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

5th Edition
EPP + 1 other
ISBN: 9781337694193
Textbook Problem
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# Refer to the definition of symmetric difference given above. Prove each of 47-52, assuming that A, B, and C are all subsets of a universal set U. ( A Δ B ) Δ C = A Δ ( B Δ C )

To determine

To prove:

(AΔB)ΔC=AΔ(BΔC).

Explanation

Given information:

A,Band C be any sets.

Concept used:

AΔB=(AB)(BA)

Calculation:

The symmetric difference of the sets Aand B is defined as,

AΔB=(AB)(BA)

Since

(AB)(BA)=(ABC)(BAC)by set difference={A(B A C)}{BC(B A C)}by the distributive law={(AB)(A A C)}{( B CB)( B C A C)}by the distributive law

={(AB)U}{U( B C A C)} by the complement laws=(AB)(BA)Cby De Morgan's law=(AB)(AB)by the set difference law

Therefore,

(AB)(BA)=(AB)(AB).......(1)

Consider,

(AΔB)ΔC={(AB)(BA)}ΔCby the definition of Δ.={(AB)(AB)}ΔCfrom equation (1)

={(AB)( AB)C}ΔCby the set difference law.={{( AB) ( AB )C}C}{C{( AB) ( AB )C}}by the definition of Δ.

={{( AB) ( AB )C}C}{{( AB) ( AB )C}C}from equation (1)={{( AB) ( AB )C}C}{{( AB) ( AB ) C}C}Cby the set difference law

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