How do you write a contrapositive

A conditional statement is logically equivalent to its contrapositive. Suppose a conditional statement of the form "If p then q " is given. The converse is "If q then p. A conditional statement is not logically equivalent to its converse. A conditional statement is not logically equivalent to its inverse.

Biconditional iff :. The biconditional of p and q is " p if, and only if, q " and is denoted p q. It is true if both p and q have the same truth values and is false if p and q have opposite truth values. Sufficient condition:. 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. Which of the other statements have to be true as well? 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.

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. Actively scan device characteristics for identification. Use precise geolocation data. Select personalised content.

Create a personalised content profile. Measure ad performance. Select basic ads. Create a personalised ads profile. Given an if-then statement "if p , then q ," we can create three related statements:. To form the converse of the conditional statement, interchange the hypothesis and the conclusion.

The converse of "If it rains, then they cancel school" is "If they cancel school, then it rains. To form the inverse of the conditional statement, take the negation of both the hypothesis and the conclusion. To form the contrapositive of the conditional statement, interchange the hypothesis and the conclusion of the inverse statement. The contrapositive of "If it rains, then they cancel school" is "If they do not cancel school, then it does not rain.

If the statement is true, then the contrapositive is also logically true. If the converse is true, then the inverse is also logically true. In the above example, since the hypothesis and conclusion are equivalent, all four statements are true. But this will not always be the case!