Show your working and give full explanations for all questions.
(1) Use truth tables to determine whether ¬(p ∨¬(p → q)) is a tautology, a contradiction, or neither
(2) Use truth tables or laws of logic to determine whether (a∧¬b) → (b∨c) is logically equivalent to ¬a ∨ b ∨ c.
(3) Write down the contrapositive of each of the following statements.
“If it’s a bird, then it has feathers.”
“If the input is invalid, then the function doesn’t return 0.”
(4) Pei Ann has been dealt two cards from a standard 52 card deck. She holds one in her left hand and one in her right.
Let p be the proposition “The card in Pei Ann’s left hand is an ace”.
Let q be the proposition “The card in Pei Ann’s right hand is an ace”.
Let r be the proposition “The card in Pei Ann’s left hand is a club”.
Let s be the proposition “The card in Pei Ann’s right hand is a club”.
Write propositions (using just p,q,r,s and logical connectives) corresponding to the following statements.
– Pei Ann has exactly one club.
– If Pei Ann has the ace of clubs in her left hand, then she doesn’t have a club in her right hand.
– Pei Ann has at least one ace that is not a club.
(If you’re not familiar with a standard 52 card deck of playing cards, have a look at http://en.wikipedia.org/wiki/Standard 52-card deck.)
(5) Show that (¬p ∨¬q) →¬q ≡ p ∨¬q using laws of logic.
