Consider the conditional statement “If 𝐴, then 𝐵,” where the hypothesis 𝐴 is “𝑥 and 𝑦 are even numbers” and the conclusion 𝐵 is “𝑥 plus 𝑦 is even.” Complete the table to give the truth value of the conditional statement and its converse, inverse, and contrapositive.
One way to test truth value is to see if you can give a counterexample, is to see if you can give a place where that statement is not true. If we find even one counterexample, the truth value of the statement is false.
We’ll start here, if 𝐴, then 𝐵. 𝑥 and 𝑦 are even. Then, 𝑥 plus 𝑦 is even. Two plus two equals four is an example. Is there any time that two even numbers would add up to an odd number? Four plus four equals eight. 10 plus 10 equals 20. When we add two even numbers, we always get an even number. The truth value for “if 𝐴, then 𝐵” is true.
Now we’re moving on, if 𝐵, then 𝐴. 𝑥 plus 𝑦 is even. Therefore, 𝑥 and 𝑦 are even numbers. Can you think of any example where 𝑥 plus 𝑦 is an even number but the two values that you add together are not even? For example, six is an even number. We can add three plus three together to equal six. In this case, the first statement is true. 𝑥 plus 𝑦 is even. But the second statement is not true. Both of the add-ins are not even. And that makes the converse, if 𝐵, then 𝐴, false.
Next one, if not 𝐴, then not 𝐵. If not 𝐴, that means 𝑥 and 𝑦 are not even. Then, 𝑥 plus 𝑦 is not even. We’re looking for any counterexamples, any places where this would not be true. 𝑥 and 𝑦 are not even. Let’s start there. We have three plus five. So three is not even. Five is not even. Then, 𝑥 plus 𝑦 equals eight. But eight is an even number. We have a case where 𝑥 and 𝑦 are not even. But that does not mean that 𝑥 plus 𝑦 is not even because it is. There’s a counterexample, which means the truth value of that statement is false. The inverse of if 𝐴, then 𝐵 is false.
Our last statement, the contrapositive, if not 𝐵, then not 𝐴. Not 𝐵 would mean 𝑥 plus 𝑦 is not even. Not 𝐴, and that means 𝑥 plus 𝑦 are not even. So we need a not even value for 𝑥 plus 𝑦. We’ll say that 𝑥 plus 𝑦 equals nine. Not 𝐴 means that 𝑥 and 𝑦 are not even.
We could say one plus eight. And one of those values, eight, is even. But both of them are not. 𝑥 and 𝑦 are not even. That means not 𝐵 is true and not 𝐴 is also true so far. And here’s the question here. Can two even numbers ever add up to an odd number? No. Since two even numbers can never add up to an odd number, the contrapositive of this statement is true, if not 𝐵, then not 𝐴.
Our table is complete with true, false, false, true.