Conditional Statements

 Introduction

The most important way to combine two statements is by implication. The implication of two statements c and d takes the form, "if f , then g ." The result of implication is called a conditional statement. It is symbolized by placing an arrow between the two letters symbolizing the two statements, as so:

f ⇒ g

Conditional statements don't necessarily imply cause and effect. They simply state that if one event happens, then another will happen. Much of geometry can be explained using conditional statements, and it is important to understand them. For example, "if a polygon has three sides, then it is a triangle" is a conditional statement.

A conditional statement has two parts, the hypothesis and the conclusion. The hypothesis is the "if" clause of the statement. It is the condition necessary for the conclusion to occur. The conclusion is the "then" clause of the statement. The conclusion is true every time the hypothesis is true. In the statement "If Julie runs fast, then she will win the race", the hypothesis is "Julie runs fast" and the conclusion is "she will win the race."

Many different statements can be made by switching the hypothesis with the conclusion and using the negation of a statement instead of the original statement. In the next section, we'll look at some conditional statements with their parts changed in certain ways, and we'll explore the truth values of such statements.

Explanation

An if-then statement or conditional statement is a statement formed when one thing implies another, but not necessarily the other way around.

In order to understand contrapositives on a mathematical level, we need to know about implication. Implication is a way to phrase if-then statements to indicate that one thing implies another. It is represented by an arrow, here typeset as "→". A statement using this arrow is known as a conditional statement, because the truth of the second value is conditional on the truth of the first. Not having electricity implies not being able to use your computer. Implication statements are only false when the first condition is true and the second condition is false. If both are true, it holds. If the first condition is false, the statement is considered vacuously true no matter what the second condition is.

Negation is used to show that a condition is not true (also known as false), and is indicated by the " ~" (tilde) symbol in front of the variable. Whether a given condition is true or false is known as its truth value. Note that a condition, represented by a variable, cannot be true and false at the same time. Anything that is always false is called a fallacy. Something that is always true is called a tautology.

• Implication - When one condition is deducible based on another. Can be written p → q, and pronounced If P then Q.
• Truth Table - A way of visually representing a conditional for all values of the variables in that conditional.
• Contrapositive - The conditional created when negating both sides of an implication. Can be written ~ q → ~ p, and said "If not Q then not P".
• Fallacy - Something that is always false.
• Tautology - Something that is always true.