Exercise 1.4.1: Proving tautologies and contradictions. About Show whether each logical expression is a tautology, contradiction or neither.

(a) (p q) (q p)

(b) (p q) (p q)

(c) (p q) p

(d) (p q) p

(e) (p q) (p q)

(f) (p q) (p q)

Exercise 1.4.2: Truth tables to prove logical equivalence. About Use truth tables to show that the following pairs of expressions are logically equivalent.

(a) p q and (p q) (q p)

(b) (p q) and p q

(c) p q and p q

Exercise 1.4.3: Proving two logical expressions are not logically equivalent. About Prove that the following pairs of expressions are not logically equivalent.

(a) p q and q p

(b) p q and p q

(c) (p q) (r q) and (p r) q

(d) p (p q) and p q

Exercise 1.4.4: Proving whether two logical expressions are equivalent. About Determine whether the following pairs of expressions are logically equivalent. Prove your answer. If the pair is logically equivalent, then use a truth table to prove your answer.

(a) (p q) and p q

(b) (p q) and p q

(c) p (p q) and p q

(d) p (p q) and p q

Exercise 1.4.5: Logical equivalence of two English statements. About Define the following propositions: j: Sally got the job. l: Sally was late for her interview r: Sally updated her resume. Express each pair of sentences using logical expressions. Then prove whether the two expressions are logically equivalent.

(a) If Sally did not get the job, then she was late for her interview or did not update her resume. If Sally updated her resume and did not get the job, then she was late for her interview.

(b) If Sally did not get the job, then she was late for her interview or did not update her resume. If Sally updated her resume and was not late for her interview, then she got the job.

(c) If Sally got the job then she was not late for her interview. If Sally did not get the job, then she was late for her interview.

(d) If Sally updated her resume or she was not late for her interview, then she got the job. If Sally got the job, then she updated her resume and was not late for her interview.

Exercise 1.4.6: Applying De Morgan's laws. About Translate each English sentence into a logical expression using the propositional variables defined below. Then negate the entire logical expression using parentheses and the negation operation. Apply De Morgan's law to the resulting expression and translate the final logical expression back into English. p: the applicant has written permission from his parents e: the applicant is at least 18 years old s: the applicant is at least 16 years old

(a) The applicant has written permission from his parents and is at least 16 years old.

(b) The applicant has written permission from his parents or is at least 18 years o