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) |