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CS/Math 211 HOMEWORK 1 Name Edit this document or print and upload a pic/pdf. 1. Fill in the truth table: p q P (q + p) + 7p (la p) ap) 2. Write a chain of equivalences to simplify (( p) + p) as much as possible. Some laws that may be useful are given in the box. Each step in your chain will use exactly one of the laws, except if you need commutativity (the last two laws) you may combine steps (e.g. false V p = p rather than false v p= p v false = p). You don't need to say the name of any law. A + B JA VB 19ASA -(A v B) = -AA-B -A AB) = -A V-B (A - B) = AAB ( AB) v = (A) ^ (BV (A v B) ACE (AACv (BA JA VA= true JA AAE false Av true = true Av false = A AA true EA A false = false AVB = BVA AABE BAA (A^ B) V A = A (A v B) AAEA 3. When a proposition is logically equivalent to true, we call it a tautology. When it's logically equivalent to false, we say it's a contradiction. If the proposition is neither a tautology nor a contradiction, we call it a contingency. Write "Tautology", "Contradiction" or "Contingency" next to the given proposition as appropriate. No need to justify your answer. qAppva b. PV (9 p) c. (9-79) 1-4 d. zp 17 (pvq) 4. Write the contrapositive of "If you don't live in the Bay Area then you don't live in California (Not a true statement of course.) 5. Recall, A logically implies B (written A=B) means whenever the A is true, B must also be true. For example, p q = pvq, since any row in a truth table that makes p^q true also makes p v q true. Fill in answers below (i.e. write "Yes" or "No"). The first row is done for you as an example. A B A=B? BA? png PVA Yes No 9 =P 79P 09 qVP -(p^9)