Law of Contrapositive

How to use the law of contrapositive to find the statement that is always true: the law, 2 examples, and their solutions.

Law

Law of Contrapositive

A conditional [p → q] and its contrapositive [~q → ~p]
have the same truth values.

If [p → q] is true, then [~q → ~p] is true.
If [p → q] is false, then [~q → ~p] is false.

This is the law of contrapositive.

Relationship between Conditional, Inverse, Converse, and Contrapositive

This picture shows the relationship between
a conditional and its inverse, converse, and contrapositive.

As you can see,
the inverse [~p → ~q] and the converse [q → p]
are the contrapositive of each other.

So, by the law of contrapositive,
the inverse and the converse
also have the same truth value.

Example 1

Example

Solution

The given statement is a conditional statement.

So the statement behind if is p:
it's raining.

And the statement behind then is q:
I'm staying home.

It says
the given statement p → q is true.

Then, by the law of contrapositive,
the contrapositive, ~q → ~p, is also true.

Write the contrapositive ~q → ~p.

If, not q, I'm not staying home,

then, not p, it's not raining.

So the contrapositive of the given statement,

if I'm not staying home,
then it's not raining,

is always true.

Example 2

Example

Solution

The given statement is a conditional statement.

So the statement behind if is p:
it's raining.

And the statement behind then is q:
I'm staying home.

The given statement is p → q.

And it says
the inverse, ~p → ~q, is true.

Then, by the law of contrapositive,
the contrapositive of the inverse,
the converse q → p,
is also true.

Write the converse q → p.

If, q, I'm staying home,

then, p, it's raining.

So the converse of the given statement,

if I'm staying home,
then it's raining,

is always true.