Converse and Inverse

We will now discuss converse, inverse, and contrapositive statements. These sound hard but are actually quite easy once you memorize what they are.

Converse

The converse of a statement is simply taking the variables in the statement and switching their place. So taking the following example:

If A then B or A -> B

The converse would be

If B then A or B -> A

I told you it was easy.

Inverse

The inverse of a statement is taking the negation of each variable so with the last example we would have the original statement of:

If A then B or A -> B

The inverse would be

If ~A then ~B or ~A -> ~B

Contrapositive

The final one is contrapositive which is taking the negation of all the variables in the converse of the statement. So the example would be:

Our original statement:

If A then B or A -> B

Our Converse

If B then A or B -> A

Finally our contrapositive

If ~B then ~A or ~B -> ~A

Truth Tables

Now lets take a look at the truth table for all these and we’ll see something interesting:

Variable 1Variable 2Original StatementConverseInverseContra Positive
ABA -> B B -> A~A -> ~ B~B -> ~ A
TTTTTT
TFFTTF
FTTFFT
FFTTTT

Notice anything about the corresponding truth tables for each statement? Well if you look closely you’ll see that the original statement has the same truths as the contrapositive. Also the converse has the same truth table as the inverse. So if a statement is true then you know that the contrapositive statement is also true. Same goes for converse and inverse statements.




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