Discrete Mathematics: An Active Approach to Mathematical Reasoning

We want to be able to expand our ability to use logic with mathematical statements. Many mathematical statements use variables. We say that an expression such as $x+y=5$ is not a statement. However, if we can give more information about $x$ and $y\text{,}$ then we can use such an expression in a statement. For example, we could say there exists an $x$ and $y$ such that $x+y=5\text{.}$ Or we could say for every $x$ there exists a $y$ with $x+y=5\text{.}$ To use statements with “for every” and “there exists,” we will need to be able to understand logical statements with quantifiers.