Question

For each of the statements below, state whether it is true or false. If true, explain why each of its directions → and ← is true. If false state which direction is false and give a counterexample.

(a) ∀x (A(x) ∨ B(x)) ↔ ∀xA(x) ∨ ∀xB(x)

(b) ∀x (A(x) ∧ B(x)) ↔ ∀xA(x) ∧ ∀xB(x)

(c) ∃x (A(x) ∨ B(x)) ↔ ∃xA(x) ∨ ∃xB(x)

(d) ∃x (A(x) ∧ B(x)) ↔ ∃xA(x) ∧ ∃xB(x)

Answer 1

(a) Only the left direction holds in this statement. Reason-> Assume that either everything has the property of A or it has properties of B. In both cases, everything has either property A or property B.

The counterexample for going against the right direction, take integers as the domain and let A be the even integers and B be the odd integers.

(b) Both directions hold true. Reason-> If everything has properties of both the A and B, then it is evident that everything has properties of A and everything has properties of B and vice versa.

(c) Both directions hold true. Reason for right-> Let there is something which has either property of A or B, then if that something has property A, then it is with property A, if that something has property B, then it is with property B.

Reason for left-> let there is something with property A or property B, in either case, that something has with the property A or B.

(d) only the right direction holds. Reason-> If there is something with property A and B then clearly there is something with property A and something aswell with property B.

The counterexample for going against the left direction, take integers as the domain and let A be the even integers and B be the odd integers.