Definitions in Proofs

Reference > Mathematics > Introduction to Proofs

We've looked at properties, but in Algebra, we don't just have properties; we also have things called definitions. A definition is a way of clarifying what something is, and definitions can be used in proofs as well. Here's an example of a definition:

Definition of Divisible: An integer n is divisible by an integer m if

is also an integer.

Does that definition make sense to you? If we say, for example, that 12 is divisible by 3, then we're saying that

is an integer. Similarly 12 is not divisible by 5 because

is not an integer.

Here's another definition:

Definition of Even: An integer n is even if it is divisible by 2.

How can we use these definitions in a proof? Here's an example:

Prove: If x + 2y = 28, and y is an integer, then x is even.

STATEMENT                                     REASON
1. x + 2y = 28                                Given
2. y is an integer                            Given
3. x + 2y - 2y = 28 - 2y                      Subtraction property of equality (1)
4. x = 28 - 2y                                Combining like terms
5. x = 2(14 - y)                              Distributive property
6. x
2 = 14 - y                                 Division property of equality
7. 14 - y is an integer                       Integers are closed under subtraction (2) *
8. x is divisible by 2                        Definition of divisible (6, 7)
8. ∴ x is even                                Definition of even

* This property is a fancy way of saying that when you subtract one integer from another, the result is always an integer.

Algebra is filled with definitions, and Geometry even more so! In Geometry you can't open your textbook without seeing a definition. Getting used to seeing definitions, and figuring out how to make use of them in proofs is going to be very important when you study Geometry!