Conditional Statements

Two common types of statements found in the study of logic are conditional and biconditional statements. They are formed by combining two statements which we then we call compound statements. What if we were to say, "If it snows, then we don't go outside." This is two statements combined. They are often called if-then statements. As in, "IF it snows, THEN we don't go outside." They are a fundamental building block of computer programming.

Writing conditional statements

A statement written in if-then format is a conditional statement.

It looks like

$p\to q$

This represents the conditional statement:

"If p then q."

A conditional statement is also called an implication.

Example 1

If a closed shape has three sides, then it is a triangle.

The part of the statement that follows the "if" is called the hypothesis. The part of the statement that follows the "then" is the conclusion.

So in the above statement,

If a closed shape has three sides, (this is the hypothesis)

Then it is a triangle. (this is the conclusion)

Example 2

Identify the hypothesis and conclusion of the following conditional statement.

A polygon is a hexagon if it has six sides.

Hypothesis: The polygon has six sides.

Conclusion: It is a hexagon.

The hypothesis does not always come first in a conditional statement. You must read it carefully to determine which part of the statement is the hypothesis and which part is the conclusion.

Truth table for conditional statement

The truth table for any two given inputs, say $A$ and $B$ , is given by:

• If $A$ and $B$ are both true, then $A\to B$ is true.
• If $A$ is true and $B$ is false, then $A\to B$ is false.
• If $A$ is false and $B$ is true, then $A\to B$ is true.
• If $A$ and $B$ are both false, then $A\to B$ is true.

Take our conditional statement that if it snows, we do not go outside.

If it is snowing ( $A$ is true) and we do go outside ( $B$ is false), then the statement $A\to B$ is false.

If it is not snowing ($A$ is false), it doesn't matter if we go outside or not ($B$ is true or false), because $A\to B$ is impossible to determine if A is false, so the statement $A\to B$ can still be true.

Biconditional statements

A biconditional statement is a combination of a statement and its opposite written in the format of "if and only if."

For example, "Two line segments are congruent if and only if they are the same length."

This is a combination of two conditional statements.

"Two line segments are congruent if they are the same length."

and

"Two line segments are the same length if they are congruent."

A biconditional statement is true if and only if both the conditional statements are true.

Biconditional statements are represented by the symbol:

$p\leftrightarrow q$

means that

$A\to B$

and

$p\to q$

That is,

$p\leftrightarrow q=\left(p\to q\right)\wedge \left(q\to p\right)$

Writing biconditional statements

Example 3

Write the two conditional statements that make up this biconditional statement:

I am punctual if and only if I am on time to school every day.

The two conditional statements that have to be true to make this statement true are:

• I am punctual if I am on time to school every day.
• I am on time to school every day if I am punctual.

Example 4

Write the two conditional statements that make up this biconditional statement:

A rectangle is a square if and only if the adjacent sides are congruent.

The two conditional statements that have to be true to make this statement true are:

• If the adjacent sides of a rectangle are congruent then it is a square.
• If a rectangle is a square then the adjacent sides are congruent.

Topics related to the Conditional Statements

Conjunction

Counter Example

Biconditional Statement

Flashcards covering the Conditional Statements

Symbolic Logic Flashcards

Introduction to Proofs Flashcards

Practice tests covering the Conditional Statements

Introduction to Proofs Practice Tests

Get help learning about conditional statements

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