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)