# Expressions Versus Sentences

Click here for a diagram that summarizes the ideas in this section.

People sometimes have trouble understanding mathematical ideas: not necessarily because the ideas are difficult, but because they are being presented in a foreign language—the language of mathematics.

The language of mathematics makes it easy to express the kinds of thoughts that mathematicians like to express. It is:

Every language has its vocabulary (the words) and its rules for combining these words into complete thoughts (the sentences). Mathematics is no exception.

As a first step in studying the mathematical language, we will make a very broad classification between the ‘nouns’ of mathematics (used to name mathematical objects of interest) and the ‘sentences’ of mathematics (which state complete mathematical thoughts).

*expression*is the mathematical analogue of an English noun; it is a correct arrangement of mathematical symbols used to represent a mathematical object of interest.

An expression does *not* state a complete thought;
it does not make sense to ask if an expression is *true* or *false*.

The most common expression types are
*numbers*, *sets*, and *functions*.

Numbers have lots of different names: for example, the expressions

all *look* different, but are all just different *names*
for the same number.
This simple idea—that numbers have lots of different names—is extremely
important in mathematics!

*sentence*is the analogue of an English sentence; it is a correct arrangement of mathematical symbols that states a complete thought.

Sentences have verbs. In the mathematical sentence ‘$\,3 + 4 = 7\,$’ , the verb is ‘$\,=\,$’.

A sentence can be (always) true, (always) false, or sometimes true/sometimes false.

For example, the sentence ‘$1 + 2 = 3$’ is true.

The sentence ‘$1 + 2 = 4$’ is false.

The sentence ‘$x = 2$’ is sometimes true/sometimes false:
it is true when $\,x\,$ is $\,2\,$,
and false otherwise.

The sentence ‘$x + 3 = 3 + x$’ is (always) true,
no matter what number is chosen for $\,x\,$.

## Examples

So, $\,x\,$ is to mathematics as *cat* is to English:
hence the title of the book,