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American Government Biology College Mathematics General CLEP General Tips Human Growth And Development Humanities Introductory Psychology Principles Of Marketing Sociology US History 1 US History 2Converse and Inverse
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 1 | Variable 2 | Original Statement | Converse | Inverse | Contra Positive |
|---|---|---|---|---|---|
| A | B | A -> B | B -> A | ~A -> ~ B | ~B -> ~ A |
| T | T | T | T | T | T |
| T | F | F | T | T | F |
| F | T | T | F | F | T |
| F | F | T | T | T | T |
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.