Department of Mathematics

Logic and Mathematical Statements

Worked Examples

Some simple equivalences

Sometimes in mathematics it is useful to replace one statement with a different, but equivalent one. Take for example the statement "If $n$ is even, then $\frac{n}{2}$ is an integer." An equivalent statement is "If $\frac{n}{2}$ is not an integer, then $n$ is not even." The original statement had the form "If A, then B" and the second one had the form "If not B, then not A." (Here A is the statement "$n$ is even", so "not A" is the statement "$n$ is not even", and B is the statement "$\frac{n}{2}$ is an integer" so "not B" is the statement "$\frac{n}{2}$ is not an integer.")

This is an often useful equivalence: "If A, then B" is equivalent to the statement "If not B, then not A".

We saw in the last section that negation of the statement "If A, then B" is the equivalent to the statement "A and not B".

Example. Consider the statement "If all rich people are happy, then all poor people are sad." Write down an equivalent statement of the form "If Not B, then Not A".

Letting A be the statement "All rich people are happy" and B be the statement "All poor people are sad." we get:

"If there exists a poor person who is not sad, then there exists a rich person who is not happy."