Write the converse, inverse, and contrapositive of the statements. (a)     ~p → ~q Converse: p → ~q~q → p     ~q → ~pq → pp → q Inverse: p → ~q~q → p     ~q → ~pq → pp → q Contrapositive: p → ~q~q → p     ~q → ~pq → pp → q (b)     ~t → s Converse: t → ~s~s → t     t → ss → ~t~s → ~t Inverse: t → ~s~s → t     t → ss → ~t~s → ~t Contrapositive: t → ~s~s → t     t → ss → ~t~s → ~t

Write each of the following conditional statements and the corresponding converse, inverse, and contrapositive as English sentences in the form “if ?, then ?.”Example: Whenever it rains, it pours. Answer: Original statement: If it rains, then it pours. Converse: If it pours, then it rains. Inverse: If it doesn’t rain, then it doesn’t pour. Contrapositive: If it doesn’t pour, then it doesn’t rain. 1.1 You can do it if you put your mind to it. 1.2 Being cold implies that you need a jacket. 1.3 Petting the cat is sufficient for it to purr.

Proofs: Construct proofs to show that the following symbolic arguments are valid.  Commas mark the breaks between premises, ‘∴’ precedes the conclusion. You may use any rule, but you must use CP (H∨E)→K ∴E→K

Consider the statement: If S = Ø then P(S) = Ø. (a) Is the statement true or false? (b) State the contrapositive. Is it true or false? (c) State the converse. Is it true or false? (d) State the contrapositive of the converse. Is it true or false? (e) Consider the statement: If S = Ø then P(S) ‡ Ø. Is this statement true or false? Is this statement equivalent to any of the statements above?