# 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 A sentence is open when it is not known if it is true or false.

### Examples:

 8 is an even number is closed (it is always true) 9 is an even number is closed (it is always false) n is an even number is open (could be true or false, depending on the value of n)

In that last example:

• if n was 4 the sentence would be true,
• if n was 5 the sentence would be false,
• etc ...

But we didn't say what value n has!

So "n is an even number" may be true or false. So it is open.

## Open Sentence

So, we get this definition:

An open sentence can be either true or false depending on 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 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)