Inverses and Contrapositives
Negation
| If you want to state that something is not going to happen, then you negate the statement by inserting "not" into it. |
Statement: An angle is right
Negation: An angle is not right
In symbols, we use the symbol "~" to indicate "not".
p: An angle is right
~p: An angle is not right
Inverses
| If you negate both the hypothesis and conclusion of a conditional statement, then you have the inverse of the statement. |
For the statement:
If it is raining, then I’ll take an umbrella.
p: it is raining, q: I’ll take an umbrella
pq
The inverse is:
If it is not raining, then I will not take an umbrella.
~p
What is the inverse of the statement "If a number is even, then it is divisible by 2"?
Contrapositives
| If you negate both the hypothesis and conclusion of a converse, then you have a contrapositive. In other words, a contrapositive is formed by exchanging the hypothesis and conclusion of a statement then negating both parts. |
For the statement:
If the radius of a circle is 5, then its diameter is 10
p: the radius of the circle is 5, q: its diameter is 10
p
The contrapositive is:
If the diameter of a circle is not 10, then the radius is not 5.
~q
What is the contrapositive of the statement "If the angle measures 90 degrees, then it is a right angle"?
