PLEASE HELP ME THANKS IN ADVANCED mathematical reasoning qeu

PLEASE HELP ME!

THANKS IN ADVANCED!

mathematical reasoning qeustion!

Solution

Let us consider the set ( A x B ) ( C x D ) = ( A x B ) ( C x D)

for every p, p ( A x B ) ( C x D )

By “ definition of ” , p A x B p C × D

By “definition of x, twice” , x , y : p = ( x , y ) : x A y B x , y : p = ( x , y ) : x C y D

By “logic: merge the two quantifications-- to bring A,C and B,D together” ,

x , y : p = ( x , y ) : x A y B x C y D

By “definition of , twice -- this introducesthe sets we\'re interested in” ,

x , y : p = ( x , y ) : x A C y B D ,

By “ definition of x ”, p ( A C ) x ( B D )

So by extensionality this proves that and x \"distribute over each other\":

( A x B ) ( C x D ) = ( A C ) x ( B D )

===p(A×B)(C×D)