Our everyday lives are full of logical statements, deductive reasoning, and counterexamples. In math we may call these "theorems'' or "postulates." In normal language, we see these as "then statements'' that can be assigned certain truth values. In other words, they can be true or false. Let's dive into the world of logical equivalence and review how converse statements fit in. We’ll also define inverse statements and contrapositive statements.
Inverse and Converse Statements
A conditional statement is a statement that has "then" in it. "If it is my birthday, then I will eat ice cream." Notice there are two things that can be assigned a truth value: It is either your birthday or not, and you will either eat ice cream or you won't.
In math, this is a true conditional statement. Every time the first condition is true (it is your birthday), the second condition (eating ice cream) is also true. But what is the converse of the statement and what is its inverse?
The converse of the statement is when you change the order of the first statement. "If it is my birthday, then I will eat ice cream" becomes "If I eat ice cream, then it is my birthday."
The inverse of the statement is when you apply a negation to both statements. "If it is my birthday, then I will eat ice cream" becomes "If it is not my birthday, then I will not eat ice cream."
The contrapositive is a combination of both the inverse and the converse for the original conditional statement. "If it is my birthday, then I will eat ice cream" becomes "If I do not eat ice cream, then it is not my birthday." The contrapositive of a conditional statement flips both the order and values (not to be confused with the biconditional statement)!
Converse Statements and Beyond
If you use the letters p and q to generalize conditionals statements, here is the quick reference table using p and q to stand in for conditional statements. This is a good reminder of how the contrapositive, inverse, and converse of a statement work.
Once you are familiar with these terms, you will see them used in other math proofs such as the converse of the Pythagorean Theorem, or even in day-to-day life.