Let P(x) and Q(x) be predicates and suppose D is the domain of x. For the statement forms in the given pair, determine whether they have the same truth value for every choice of P(x), Q(x), and D, or whether there is a choice of P(x), Q(x), and D for which they have opposite truth values.
∃x ∈ D, (P(x) ∨ Q(x)) and (∃x ∈ D, P(x)) ∨ (∃x ∈ D, Q(x))
These two statements have the same truth values for every choice of P(x), Q(x), and D. If there exists an x in D such that P(x) is true, or there exists an x in D such that Q(x) is true, then it is necessarily true that there exists an x in D such that P(x) or Q(x) is true.
These two statements do not have the same truth values for every choice of P(x), Q(x), and D. For example, if we let D = R, P(x) be "x is positive," and Q(x) be "x is negative," then the first statement is false and the second statement is true.
These two statements do not have the same truth values for every choice of P(x), Q(x), and D. For example, if we let D = R, P(x) be "x is positive," and Q(x) be "x is negative," then the first statement is true and the second statement is false.
These two statements do not have the same truth values for every choice of P(x), Q(x), and D. For example, if we let D = Z, P(x) be "x is a whole number," and Q(x) be "x is a decimal," then the first statement is false and the second statement is true.
These two statements do not have the same truth values for every choice of P(x), Q(x), and D. For example, if we let D = Z, P(x) be "x is a whole number," and Q(x) be "x is a decimal," then the first statement is true and the second statement is false.