IF ... THEN ... : A Very Common Structure in Mathematics

The most common and basic structure used in mathematical language is the

"If (some hypothesis) then (some conclusion)"

sentence. Almost every result in mathematics can be stated using one or more "If ... then ... " sentences.

"If A then B" means that when the hypothesis A is true, then the conclusion B must also be true.

If the hypothesis is false, then the "If ... then ... " sentence makes no claim about the truth or falsity of the conclusion - the conclusion may be either true or false.

Even in everyday life you have probably encountered "If ... then ..". statements for a long time. A parent might try to encourage a child with a statement such as " If you clean your room then I will buy you an ice cream cone".

To show that an "If . . . then . . . " statement is not valid (not true), all we need to do is find a single example where the hypothesis is true and the conclusion is false. Such an example with a true hypothesis and false conclusion is called a counterexample for the "If . . . then . . . " statement. A valid "If . . . then . . . " statement has no counterexample.

A counterexample to the statement "If A then B" is an example in which A is true and B is false.

The only way for the statement " If you clean your room then I will buy you an ice cream cone" to be false is if the child cleaned the room and the parent did not buy the ice cream cone. If the child did not clean the room but the parent still bought the ice cream cone we would say that the statement was true.

The statement "If n is a positive integer, then n^2 + 5n + 5 is a prime number" has hypotheses "n is a positive integer" and conclusion "n^2 + 5n + 5 is a prime number". This "If ... the" statement is false since replacing n with the number 5 will make the hypothesis true and the conclusion false. The number 5 is a counterexample for the statement. Every invalid "If . . . then . . . " statement has at least one counterexample, and the most convincing way to show that a statement is not valid is to find a counterexample to the statement.

A number of other language structures can be translated into the "If ... then ..". form. The statements below all mean the same as "If (A) then (B)" :

"All (A) are (B)". Every (A) is (B)". "Each (A) is (B)".
"Whenever (A), then (B)". "(B) whenever (A)". "(A) only if (B)".
"(A) implies (B)". "(A) ⇒ (B)" (the symbol " ⇒ " means "implies" )

Practice 3: Restate "If (a shape is a square) then (the shape is a rectangle)" as many ways as you can.

"If ... then ... " statements occur hundreds of times in every mathematics book, including this one. It is important that you are able to recognize the various forms of "If ... then ... " statements and that you are able to distinguish the hypotheses from the conclusions.