What is Converse, Inverse, and Contrapositive
In geometry, many declarations are written in conditional form "If ...., then....." For Example: "If two angles are right angles, then they are congruent." The if statement is called the hypothesis and the then statement is called the conclusion or consequent. Every conditional statement "if p then q" (symbolic notation p/q) has three other conditionals associated with it:
1. Converse (if q then p) (q/p)
2. Inverse (if not p then not q) (~p/~q)
3. Contrapositive (if not q then not p) (~q/~p)
For the statement about right angles, the three related statements are:
1. Converse: If two angles are congruent, then they are right angles.
2. Inverse: If it is not true that two angles are right angles, then they are not congruent angles.
3. Contrapositive: If it is not true that two angles are congruent, then they are not both right angles.
Theorem 1 : If a conditional statement is true, then the contrapositive of the statement is true. A statement and its contrapositive are logically equivalent.
*** Be careful: If a statement is true, its converse or inverse is not necessarily true. See if the converse of this statement is true: "An equilateral triangle is equiangular."
See if the inverse of this statement is true: "An equilateral polygon is equiangular."
If and only if (iff)
When both a statement and its converse are true, there is a convenient way to combine the two into one. It is by means of the phrase "if and only if." When we say "p if and only if q", we mean both "if p then q and if q then p." We can represent the phrase "if and only if" by the symbol /. To write p / q means that both p / q and q / p are true. Mathematicians usually abbreviate the phase "if and only if " by writing "iff."
Sometimes when we prove a theorem, we use the chain rule.
Definition 1
The chain rule is a pattern of reasoning represented as follows: Whenever p/q is true and q/r is true, we can conclude p/r is true.
Example 1
"if A=B then B=C; if B=C then C=D"
If both statements are true, according to the chain rule we conclude "if A=B then C=D".