Question

(1) Determine whether the propositions p → (q ∨ ¬r) and (p ∧ ¬q) → ¬r are logically equivalent using either a truth table or laws of logic.

(2) Let A, B and C be sets. If a is the proposition “x ∈ A”, b is the proposition “x ∈ B” and
c is the proposition “x ∈ C”, write down a proposition involving a, b and c that is logically equivalentto“x∈A∪(B−C)”.

(3) Consider the statement ∀x∃y¬P(x,y). Write down a negation of the statement that does not use the symbol ¬.

(4) Under the interpretation where x and y are in R − {0} and P (x, y) is “xy ≥ 0”, is the original statement in (3) true or is its negation true?

(5) Is the statement ( ∃x(P (x) ∨ Q(x)) ) → ( (∃xP (x)) ∨ (∃xQ(x)) ) valid? If it is, explain why. If it isn’t, give an interpretation under which it is false.

Answer 1

where in ( AUB) if their is a single T either in A or B truth table is T.

(AB) if their is single F in either A or B truth table is F

for ~ A negation A T becomes F and F becomes T

in () for only TF combination becomes F remaining all are T.