Conditional Statements and Logic

In mathematics, logic is the foundation of proof. One of the most common logical structures is the conditional statement, often written in the "if-then" form.

A conditional statement connects a hypothesis ($p$) to a conclusion ($q$). We write this as $p \rightarrow q$, read as "If $p$, then $q$."

Related Conditional Statements

From a single conditional statement, we can create three related statements: the converse, the inverse, and the contrapositive.

Let's use an example statement: "If a triangle is equilateral, then it is isosceles."

• Hypothesis ($p$): A triangle is equilateral.
• Conclusion ($q$): It is isosceles.

1. The Converse ($q \rightarrow p$)

Swap the hypothesis and conclusion.

• Statement: "If a triangle is isosceles, then it is equilateral."
• Note: The converse of a true statement is not always true!

2. The Inverse ($\sim p \rightarrow \sim q$)

Negate both the hypothesis and the conclusion. (The symbol $\sim$ means "not").

• Statement: "If a triangle is not equilateral, then it is not isosceles."

3. The Contrapositive ($\sim q \rightarrow \sim p$)

Swap and negate both the hypothesis and the conclusion.

• Statement: "If a triangle is not isosceles, then it is not equilateral."
• Note: A conditional statement and its contrapositive are logically equivalent. If the original statement is true, the contrapositive is always true.

Disproving with Counterexamples

To prove a conditional statement is false, you do not need a long argument. You only need one counterexample—a specific case where the hypothesis is true, but the conclusion is false.

Example: Is the statement "If $x^2 > 0$, then $x > 0$" true?

• Let's test $x = -3$.
• Hypothesis: $(-3)^2 = 9 > 0$ (True).
• Conclusion: $-3 > 0$ (False).

Since we found a case where the hypothesis is true but the conclusion is false, the entire statement is false. $x = -3$ is our counterexample.

Biconditional Statements

When a conditional statement ($p \rightarrow q$) AND its converse ($q \rightarrow p$) are both true, we can combine them into a biconditional statement.

Biconditionals use the phrase "if and only if" (often abbreviated as "iff") and are written as $p \leftrightarrow q$. Good mathematical definitions are always biconditional.

• Example: "A polygon is a triangle if and only if it has exactly three sides." This means if it's a triangle, it has three sides, AND if it has three sides, it is a triangle.