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For each proposition below, let us decide whether it is true or false. Assume thedomain of variables to be , the set of integers.
The conjunction is true if and only if and In this case the disjunction is true, and hence the implication is also true. In other cases the conjunction is false, and hence the implication is true. Therefore, for all integers and the statement ((x = 5) ∧ (y = 1)) → ((x > 10) ∨ (y > 0)) is true.
Since for each integer the statement is true, we conclude the disjunction is also true, and hence the statement is true.
(3) for all predicates
Since the statements and have the same values, and hence the equivalence is true.