# 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

## 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.