Let A be nonepty set and P(A) be the power set of A. Recall the definition of power set: P(A)={x|x sube A} Show that symmetric deference operation on

Bergen 2021-02-05 Answered
Let A be nonepty set and P(A) be the power set of A. Recall the definition of power set:
P(A)={xxA}
Show that symmetric deference operation on P(A) define by the formula
xy=(xyc)(yxc),xP(A),yp(A)
(where yc is the complement of y) the following statement istrue:
The algebraic operation o+ is commutative and associative on P(A).
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Expert Answer

Derrick
Answered 2021-02-06 Author has 94 answers
Commutative:
For xP(A)andyP(A),
xy=(xyc)(yxc)and
yx=(yxc)(xyc)
=(xyc)(yxc)
=xy
Associative
For x,yandzP(A)
(xy)z=((xy)zc)(zn(xy)c)
=(z((xyc)(yxc)c)(((xyc)(yxc))zc)
=(z(xcy)(ycx))((xyc)zc)((yxc)zc)
=(z(xc(ycx))((yyc)x))(((xyc)(yxc))zc)
=(z((xcyc)(xcx))((yyc)(yx)))(((xy)zc)((yxc)zc))
=(z((xcyc)(xcx))u((yyc)(yx)))(xyczc)(yxczc)

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