key2sp00 - COT3100C-01, Spring 2000 S. Lang Solution Key to...

Info iconThis preview shows pages 1–2. Sign up to view the full content.

View Full Document Right Arrow Icon
COT3100C-01, Spring 2000 S. Lang Solution Key to Assignment #2 (40 pts.) Feb. 11, 2000 1. (12 pts.) Prove each of the following statements in Parts (a) - (f), assuming the symbols A , B , and C represent (arbitrary) sets. Your proof must be based on the appropriate definitions and the laws and theorems listed below ( but only these ). Be sure to explain each step of your proof. (Commutative Law) A B = B A , A B = B A . (Associative Law) ( A B ) C = A ( B C ), ( A B ) C = A ( B C ). (Distributive Law) A ( B C ) = ( A B ) ( A C ) , A ( B C ) = ( A B ) ( A C ) . (Idempotent Property) A A = A , A A = A . (De Morgan’s Law) ¬ ( A B ) = ¬ A ¬ B , ¬ ( A B ) = ¬ A ¬ B . (Double Negation) ¬ ( ¬ A ) = A . (Complementary Property) A ¬ A = , A ¬ A = U , where U denotes the universe. Theorem: (a) A B A . (b) A A B . (c) A = . (d) A = A . (e) A B iff ¬ B ¬ A (f) A B iff A ¬ B = . (g) A - B = A ¬ B . (h) ( A B and B C ) implies A C . (i) A B implies A C B C . (j) A B implies A C B C . (a) If A B = C , then C A and C B . Since C = A B ---- (1) by assumption, and A B A ---- (2) by the above theorem (a), so substituting (1) into (2) proves C A . Similarly, since A B = B A B ---- (3) by the commutative law and by the above theorem (a), substituting (1) into (3) yields C B . (b)
Background image of page 1

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
Image of page 2
This is the end of the preview. Sign up to access the rest of the document.

Page1 / 3

key2sp00 - COT3100C-01, Spring 2000 S. Lang Solution Key to...

This preview shows document pages 1 - 2. Sign up to view the full document.

View Full Document Right Arrow Icon
Ask a homework question - tutors are online