discrete structures

1. Show that ( p → q ) ᴧ ( p → r ) and p → ( q ᴧ r ) are logically equivalent.

2. Determine whether (¬ q ᴧ ( p → q ) ) → ¬ p is a tautology.

3. Express the statement “If a person is female and is a parent, then this person is someone’s mother “ as a logical expression involving predicates, quantifiers with a universe of discourse consisting of all people and logical connectives.

4. Prove that the product of two rational numbers is rational.

5. Let A, B, and C be sets. Show that ( BA ) U ( C – A ) = ( B U C ) – A.

6. Let G and H be subsets of a universal set U. Show that G H if and only if .

7. Find the product AB, where, A = , B = .

8. Draw graph of the function: f(x) = ⌈ 1 / x ⌉

9. Show that if ab (mod m) and c d (mod m), where a, b, c, d, and m are integers with m 2, then a – c b – d (mod m).

10. Compute each of these double sums,

(a) (b)