1Let the universe of discourse be the set of relations on th

1)Let the universe of discourse be the set of relations on the set A. Write the following English sentences as a logical expression, using quantifers, logical expressions, and setnotation.

(a) A symmetric relation on a set A is not necessarily reexive.

(b) Every reexive relation is anti-symmetric.

2)Let A, B, C , and D be subsets of a universal set U

.(a) Prove that if A B, then (complement)B (complement)A

3)Let f  : X   Y  and g : Y  Z  be one-to-one correspondences.

(a) Prove that gf must be a one-to-one correspondence

(b) Prove that (gf )^1= f^ 1g^1

Solution