November 10, 2008

This resource has been contributed by Winpossible, and can also be accessed on their website by clicking here - Converse, Inverse, Biconditional and Contrapositive.

In this mini-lesson you'll learn with the help of some examples, the definition and basics of converse, inverse, biconditional and contrapositive of a conditional. This might seem complicated here in text, but once you have instructor explain it to you in their voice and handwriting in the video, it will be fairly easy for you to understand.

The converse of a conditional is formed by interchanging the hypothesis and the conclusion.

Statement: if*p*, then *q*

Converse: if*q*, then *p*

i.e.*q* ? *p*. E.g. for the conditional “If a polygon is a hexagon, then the polygon has exactly six sides”, the converse is “If a polygon has exactly six sides, then the polygon is a hexagon”.

The inverse of a conditional is formed by negating the hypothesis and the conclusion i.e. ~*p* ? ~*q*. E.g. the conditional, “If you serve imported sparkling water, then you have good taste” has the inverse, “If you do not serve imported sparkling water, then you do not have good taste”. It should be easy to see that the converse of the inverse is the contrapositive. For example, the converse of *p*q is *q**p* and the inverse of *p*q is ~*p*~*q*.

If a conditional and its converse are both true, they can be combined into a single statement by using the words “if and only if”. The statement that contains the words “if and only if” is called a bioconditional. Its basic form is;

*p* if and only if *q*. i.e. *p* *q*.

E.g. the biconditional statement “*x* + 5 = 12, “if and only if” *x* = 7”.

The contrapositive of a conditional is a new conditional with the hypothesis and conclusion interchanged and the hypothesis and conclusion both negated i.e. ~*p* ? ~*q*. E.g. the conditional, “if two lines are parallel, then they do not intersect” has the contrapositive, “if two lines intersect, then they are not parallel.”

In this mini-lesson you'll learn with the help of some examples, the definition and basics of converse, inverse, biconditional and contrapositive of a conditional. This might seem complicated here in text, but once you have instructor explain it to you in their voice and handwriting in the video, it will be fairly easy for you to understand.

The converse of a conditional is formed by interchanging the hypothesis and the conclusion.

Statement: if

Converse: if

i.e.

The inverse of a conditional is formed by negating the hypothesis and the conclusion i.e. ~

If a conditional and its converse are both true, they can be combined into a single statement by using the words “if and only if”. The statement that contains the words “if and only if” is called a bioconditional. Its basic form is;

E.g. the biconditional statement “

The contrapositive of a conditional is a new conditional with the hypothesis and conclusion interchanged and the hypothesis and conclusion both negated i.e. ~

This FREE mini-lesson is a part of Winpossible's online course that covers all topics within Geometry. Click on the video below to go through it. If you like it, you can buy our online course in Geometry by clicking here.

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