Kenneth Rosen Edition 7 Exercise 2.2 Question 37 (Page No. 137)

Question

Show that if $A$ is a subset of a universal set $U$, then

1. $A \oplus A = \phi.$
2. $A \oplus \phi = A.$
3. $A \oplus U = \sim A.$
4. $A \oplus \sim A= U.$

Answer 1

Symmetric difference or Boolean sum:

$A\triangle B / A\oplus B = \left\{x \ \mid x\in A  \ \text{or} \  x\in B \ \text{but}  \ x\notin A\cap B \right\}$

• $A\triangle B = A\oplus B  = (A-B)\cup(B - A) $
• $A\triangle B = A\oplus B =  (A\cup B) - (A\cap B) $

$a) \ A \oplus A = \phi.$

$\implies A\oplus A = (A - A)\cup (A -  A) = \phi \cup \phi = \phi$

$b) \ A \oplus \phi = A.$

$\implies A\oplus \phi = (A\cup\phi)  - (A\cap \phi) = A -  \phi = A$

$c) \ A \oplus U =  \ \sim A.$

$\implies A\oplus U = (A\cup U) - (A\cap U) = U - A= \  \sim A$

$d) \ A \oplus \sim A= U.$

$\implies A\oplus \sim A = (A\ \cup\sim A) -  (A \ \cap \sim A) = U - \phi = U$

Answer 2

A xor B = (A-B) union (B-A)

i. (A-A)UNION(A-A)=PHI UNION PHI= PHI . HENCE PROVED.

II. (A-PHI)UNION (PHI-A)= A UNION PHI = A

III. (A-U) UNION (U-A) = PHI UNION A' = A'

IV. (A-A') UNION (A' - A)= A UNION A' = U

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