Theorem 2.1.1 Logical Equivalences Given any statement variables p, q, and r, a tautology t and a contradiction c, the following logical equivalences hold. 1. Commutative laws: P Vq = qVp (p V q) V r = p V (q V r) p V (q ^ r) = (p v q) ^ (p v r) P^q = q^p (p^q)^r=p ^ (q ^ r) рл (qvr) %3D(рлд)v(р^г) 2. Associative laws: 3. Distributive laws: 4. Identity laws: p^t=p V c = p p 5. Negation laws: PV ~p=t P^~p = c 6. Double negative law: ~(~p) = p 7. Idempotent laws: p V p = p P^p=p 8. Universal bound laws: Pvt=t рлс3Dс 9. De Morgan's laws: ~(p ^ q) = ~p V ~q p V (p ^ q) = P (p V q) = ~p^~q p^ (p v q) = p 10. Absorption laws: 11. Negations of t and c: ~t = c ~c = t (р^(~(~руq) V (р ^Ф %3Dр

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This hypothesis always contains the statement of equality.

Use the following definitions and the laws of logic to prove the following statement. • Definition of Cartesian product: (x, y) E Ax B + (x€ A) ^ (y E B) • Definition of union: x E AUB → (x E A) V (x € B) Statement. For any sets A, B, C, A × (BUC) = (A × B) U (A × C).

What is the size of the symmetric difference of two finite sets A and B, that is AA BI ? o IIAI - |B|| O IAl +|B| – 2|A B| O IAl +|B| – |AN BỊ O IA| + |B|

Suppose A and B are two finite sets such that |B| = |A| - 1 and the number of relations on A is 512. Determine: |B|:  Number of relations from B to A: