Discrete Mathematics Show that A x B u C A x B u A x CSolut

Discrete Mathematics

Show that:
A x (B u C) = (A x B) u (A x C)

Solution

Since it is a cross product of two sets, every element of both sides is an ordered pair. Hence let arbirarily, (x,y) be an element of A x (BUC).

==> x is in A and y is in (B U C)

==> x is in A and y is either B or C or in both.

==> Taking pair wise (A,B) or (A,C)

(x,y) is either in (A x B) or in (A x C) or in both

==> (x,y) is in (A x B) U (A x C)

Hence, A x ( B U C) is a subset of (A x B ) U ( A x C ) ............. (1)

2) Now let (p,q) be another arbitray element in (A x B) U (A x C)

==> p is in A and q is in either of B or C or in both

In either case of the above, we have p is in A and q is in (B U C)

==> (p,q) is in A x (B U C)

Hence, (A x B) U (A x C) is a subset of A x (B U C) ....... (2)

Thus from (1) and (2), it is proved that

A x ( B U C) = (A x B ) U ( A x C )