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

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

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)

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

View Full Document
This is the end of the preview. Sign up to access the rest of the document.

{[ snackBarMessage ]}

### 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
Ask a homework question - tutors are online