How to construct a truth table for finding-For natural numbers m,n, is it true that mn is even iff m and n are even?
A proposition is something that either true or false. For example, "5 is odd" and "5 is even" are propositions. In contrast, the property of being odd is not a proposition; it becomes one when it is applied to a concrete number.
Truth tables are used in propositional logic, where elementary building blocks of logical statements are entire propositions. Propositional logic has no way of relating two different propositions. For example, if p is "5 is odd" and q is "7 is odd," there is no way in propositional logic to express the fact that q is obtained from p by replacing 5 with 7. From the standpoint of propositional logic, p and q are black boxes with no internal structure.
The next step in logic complexity is predicate, or first-order, logic. It has ways to describe the structure of propositions because its elementary blocks are properties, or predicates, applied to objects, e.g., numbers. In predicate logic, one can express the claim that if n is odd, then n + 2 is also odd: Odd(n) -> Odd(n + 2). It is crucial that both Odd(n) and Odd(n + 2) contain the same variable n.
Now, there is a major tradeoff between propositional and predicate logics. On the one hand, predicate logic is more expressive. For example, "mn is odd iff m and n are odd" can only be expressed in predicate logic. On the other hand, for any given statement in propositional logic, it is possible to determine if it is a tautology by constructing a truth table. In predicate logic, there is no general way to determine the truth value of a given statement.
Therefore, you can't construct a truth table for your claim. If it is false, then, as Plato showed, one can come up with a counterexample. If it is true, on the other hand, the only way to show it is to construct a proof of the statement.
If you want to construct a truth table and form an "iff" statement, you will need to use all combinations of odd/even.
In this case, we only have 3:
1. Both even. For any x,y in the naturals: Let n = 2x and m = 2y. Then nm = 2(2xy). Let z be an integer such that z = 2xy. Then, nm = 2z. This is even by definition.
2. Both odd. For any x,y in the naturals: Let n = 2x + 1 and m = 2y + 1. Then nm = 4xy + 2x + 2y + 1 = 2(2xy + x + y) + 1. Let z be an integer such that z = (2xy + x + y). Then, nm = 2z + 1. This is odd by defintion.
3. One odd, one even. Because the variables are generalized, it does not matter whether n or m is even. Let n = 2x and m = 2y + 1. Then nm = 2x(2y + 1) = 2(2xy + x). Let z be an integer such that z = 2xy + x. Then nm = 2z. This is even by definition.
So, you do NOT have the quality: For natural numbers n, m, nm is even iff n and m are even.
Your truth table should resemble the following:
n is even. m is even. nm is even.
T _______T_______T
T________F_______T
F________T_______T
F________F _______F