Mathematical Language

Two statements are equivalent if they always have the same logical value (a logical value is either "true" or "false", that is, if they are both true or are both false. The statements "$x = 3$" and "$x + 2 = 5$" are equivalent statements because if one of them is true then so is the other, and if one of them is false then so is the other. The statements "$x = 3$" and "$x^2 – 4x + 3 = 0$" are not equivalent since $x = 1$ makes the second statement true but the first one false.

The compound statement "$A$ and $B$ are true" is equivalent to "both of $A$ and $B$ are true".

If $A$ or if $B$ or if both are false, then the statement "$A$ and $B$ are true" is false. The statement "$x2 = 4$ and $x > 0$" is true when $x = 2$ and is false for every other value of $x$.

The compound statement "$A$ or $B$ is true" is equivalent to "at least one of $A$ or $B$ is true".

If both $A$ and $B$ are false, then the statement "$A$ or $B$ is true" is false. The statement "$x^2 = 4$ or $x > 0$" is true if $x = –2$ or $x$ is any positive number. The statement is false when $x = –3$ and for lots of other values of $x$.

Practice 1: Which values of $x$ make each statement true?
(a) "$x < 5$"

(b) "$x + 2 = 6$"

(c) "$x^2 – 10x + 24 = 0$"

(d) "$(a)$ and $(b)$"

(e) "$(a)$ or $(c)$"

For some simple statements we can construct the negation just by adding the word "not".

When the statement contains words such as "all", "no", or "some," then its negation is more complicated.

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

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

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.

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 $(B)$" :

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.

The statement "If $(A)$ then $(B)$" means that if the hypothesis $A$ is true, then the conclusion $B$ is guaranteed to be true.

Suppose we know that in a certain town the statement

is a true statement,. What can we validly conclude about a red building? Based on the information we have, we can validly conclude that the red building is "not a church" since every church is green. We can also conclude that a blue building is not a church. In fact, we can conclude that every "not green" building is "not a church." That is, if the conclusion of a valid "If ... then ... " statement is false, then the hypothesis must also be false.

What about a green building in this town? The green building may or may not be a church – perhaps every post office is also painted green. Or perhaps every building in town is green, in which case the statement "If (a building is a church) then (the building is green)" is certainly true.

Practice 4: Write the contrapositive form of each of the following statements.

(a) If a function is differentiable then it is continuous.

(b) All men are mortal.

(c) If ($x$ equals 3) then ($x2 – 5x + 6$ equals 0)

(d) If (2 divides $x$ and 3 divides $x$) then (6 divides $x$).

If we switch the hypotheses and the conclusion of an "If $A$ then $B$" statement we get the converse "If $B$ then $A$".

The converse of an "If ... then ... " statement is a new statement with the hypothesis and conclusion switched: the converse of "If $(A)$ then $(B)$" is "If $(B)$ then $(A)$". For example, the converse of "If (a building is a church) then (the building is green)" is "If (a building is green) then (the building is a church)". The converse of an "If ... then ... " statement is not equivalent to the original "If ... then ... " statement. The statement "If $x = 2$, then $x^2 = 4$" is true, but the converse statement "If $x^2 = 4$, then $x = 2$" is not true because $x = –2$ makes the hypothesis of the converse true and the conclusion false.

Practice 1: (a) All values of $x$ less than 5.

(b) $x = 4$

(c) Both $x = 4$ and $x = 6$.

(d) $x = 4$

(e) $x = 6$ and all $x$ less than 5.

Practice 2: (a) $x + 5 < 3$.

(b) At least one prime number is even.
There is an even prime number.

(c) $x^2 ≥ 4$.

(d) $x$ does not divide 2 or $x$ does not divide 3.

(e) At least one mathematician can sing well.
There is a mathematician who can sing well.

Practice 3: Here are several ways to restate "If (a shape is a square) then (the shape is a rectangle)".
All squares are rectangles.
Every square is a rectangle.
Each square is a rectangle.
Whenever a shape is a square, then it is a rectangle.
A shape is a rectangle whenever it is a square.
A shape is a square only if it is a rectangle.
A shape is a square implies that it is a rectangle.
Being a square implies being a rectangle.

Practice 4: (a) statement "If a function is differentiable then it is continuous".
contrapositive "If a function is not continuous then it is not differentiable".

(b) statement "All men are mortal".
contrapositves "All immortals are not men".
"If a thing is not mortal then it is not human".

(c) statement "If ($x$ equals 3) then ($x^2 – 5x + 6$ equals 0)".
contrapositive "If ($x^2 – 5x + 6$ does not equal 0) then ($x$ does not equal 3)".

(d) statement "If (2 divides $x$ and 3 divides $x$) then (6 divides $x$)".
contrapositive "If (6 does not divide $x$) then (2 does not divide $x$ or 3 does not divide $x$)".