Given an if-then statement "if p , then q ," we can create three related statements: A conditional statement consists of two parts, a hypothesis in the “if” clause and a conclusion in the “then” clause. For instance, “If it rains, then they cancel school.” To form the converse of the conditional statement, interchange the hypothesis and the conclusion. 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.
If the statement is true, then the contrapositive is also logically true. If the converse is true, then the inverse is also logically true.
Example 1:
In the above example, since the hypothesis and conclusion are equivalent, all four statements are true. But this will not always be the case!
Example 2:
Write the negation of the statement “An angle is a right angle if and only if it is of measure 90°” Let p: An angle is a right angle. q: An angle is of measure 90°. ∴ The symbolic form of the above Statement is p ↔ q. Note that negation of ‘p ↔ q’ is (p ∧ ∼q) ∨ (q ∧ ∼p). ∴ The negation of the given statement is ‘An angle is a right angle and is not of measure 90° or an angle is of measure 90° and not a right angle. Concept: Logical Connective, Simple and Compound Statements Is there an error in this question or solution? |