Open Sentences

An example of an open sentence: x + 3 = 6

First ... what is a "Sentence" ?

Just like an English sentence, in mathematics a sentence says something:

English:

The sun is shining.
Hawaii is in the Pacific Ocean.

Mathematics:

3 + 3 = 6
10 is an even number

Now ... what is a "Closed Sentence" or an "Open Sentence" ?

Closed	A closed sentence is always true (or always false).
Open	If you don't know whether a sentence is true or false, then it is open.

Examples:

10 is an even number	is closed (you know it is always true)
11 is an even number	is closed (you know it is always false)
n is an even number	is open (it may be true or false, depending on what value n is)

In that last example, if n was 2 then it would be true, if n were 3 it would be false, and so on. But we have not said what value n has, so we can't say if it is true or false.

Open Sentence

So, we get this definition:

An open sentence can be either true or false depending what values are used.

Variables

The value we don't know is called a variable (also called an unknown)

In this example of an open sentence, x is a variable:

x + 3 = 8

In this example, w and q are both variables:

w + q = 2

Solving

Solving the open sentence means finding a value for the variable that makes the sentence true.

Example: Solve x + 3 = 8

Let us subtract 3 from both sides:

x + 3 - 3 = 8 - 3

x = 5

Check: 5 + 3 = 8 is true

So we have solved x + 3 = 8 by making x = 5

Some More Examples

Here are some more examples of closed and open sentences for you:

Closed Sentences:

A square has four corners	always true
6 is less than 5	always false
-3 is a negative number	always true

Open Sentences:

A triangle has n sides	Can be true or false (depending on the value of n)
z is a positive number	Can be true or false (depending on the value of z)
3y = 4x + 2	Can be true or false (depending on the values of x and y)
a + b = c + d	Can be true or false (depending on the values of a,b,c,d)