Question

for any three sets A,B,C prove that (a-b)*c= (a*c)-(b*c)

Answer 1

Concept:

$A{\times}B=${(x,y)/x∈A, y∈B }

$A-B=${x/x∈A and x∉B }

$\textbf{To prove:}(A-B){\times}C=(A{\times}C)-(B{\times}C)$

$Let\:(x,y){\in}(A-B){\times}C$

$\iff\;x{\in}A-B\;,\;y{\in}C$

$\iff\;x{\in}A\;and\;x{\notin}B\;y{\in}C$

$\iff\;(x,y){\in}A{\times}C\;and\;(x,y){\notin}B{\times}C$

$\iff\;(x,y){\in}(A{\times}C)-(B{\times}C)$

$\therefore\bf(A-B){\times}C=(A{\times}C)-(B{\times}C)$