29 Aug 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)
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