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Conditional statements make appearances everywhere. In mathematics or elsewhere, it doesn't take long to run into something of the form “If P then Q.” Conditional statements are indeed important. What is also important are statements that are related to the original conditional statement by changing the position of P, Q and the negation of a statement. Starting with an original statement, we end up with three new conditional statements that are named the converse, the contrapositive, and the inverse.
Before we define the converse, contrapositive, and inverse of a conditional statement, we need to examine the topic of negation. Every statement in logic is either true or false. The negation of a statement simply involves the insertion of the word “not” at the proper part of the statement. The addition of the word “not” is done so that it changes the truth status of the statement.
It will help to look at an example. The statement “The right triangle is equilateral” has negation “The right triangle is not equilateral.” The negation of “10 is an even number” is the statement “10 is not an even number.” Of course, for this last example, we could use the definition of an odd number and instead say that “10 is an odd number.” We note that the truth of a statement is the opposite of that of the negation.
We will examine this idea in a more abstract setting. When the statement P is true, the statement “not P” is false. Similarly, if P is false, its negation “not P” is true. Negations are commonly denoted with a tilde ~. So instead of writing “not P” we can write ~P.
Converse, Contrapositive, and Inverse
Now we can define the converse, the contrapositive and the inverse of a conditional statement. We start with the conditional statement “If P then Q.”
- The converse of the conditional statement is “If Q then P.”
- The contrapositive of the conditional statement is “If not Q then not P.”
- The inverse of the conditional statement is “If not P then not Q.”
We will see how these statements work with an example. Suppose we start with the conditional statement “If it rained last night, then the sidewalk is wet.”
- The converse of the conditional statement is “If the sidewalk is wet, then it rained last night.”
- The contrapositive of the conditional statement is “If the sidewalk is not wet, then it did not rain last night.”
- The inverse of the conditional statement is “If it did not rain last night, then the sidewalk is not wet.”
We may wonder why it is important to form these other conditional statements from our initial one. A careful look at the above example reveals something. Suppose that the original statement “If it rained last night, then the sidewalk is wet” is true. Which of the other statements have to be true as well?
- The converse “If the sidewalk is wet, then it rained last night” is not necessarily true. The sidewalk could be wet for other reasons.
- The inverse “If it did not rain last night, then the sidewalk is not wet” is not necessarily true. Again, just because it did not rain does not mean that the sidewalk is not wet.
- The contrapositive “If the sidewalk is not wet, then it did not rain last night” is a true statement.
What we see from this example (and what can be proved mathematically) is that a conditional statement has the same truth value as its contrapositive. We say that these two statements are logically equivalent. We also see that a conditional statement is not logically equivalent to its converse and inverse.
Since a conditional statement and its contrapositive are logically equivalent, we can use this to our advantage when we are proving mathematical theorems. Rather than prove the truth of a conditional statement directly, we can instead use the indirect proof strategy of proving the truth of that statement's contrapositive. Contrapositive proofs work because if the contrapositive is true, due to logical equivalence, the original conditional statement is also true.
It turns out that even though the converse and inverse are not logically equivalent to the original conditional statement, they are logically equivalent to one another. There is an easy explanation for this. We start with the conditional statement “If Q then P”. The contrapositive of this statement is “If not P then not Q.” Since the inverse is the contrapositive of the converse, the converse and inverse are logically equivalent.