The importance of definitions

The Importance of Definitions

Any time your claim uses defined terms or notation, consider going to the definition and finding out what the terminology or notation means. Similarly, if one of your facts in the fact bank uses a defined term or notation, consider using the definition. Definitions are always available to add to your fact bank.

Example

For this example, the only numbers that we are talking about are integers.

Definition. Say that x | y if and only if x ≠ 0 and there exists an integer k so that y = kx.

For example, 5 | 15 since 15 = (3)(5). If x | y we say that x divides y, or x is a factor of y.

Definition. Say that x ≡ y (mod m) if and only if m > 0 and m | (x − y).

For example, 17 ≡ 1 (mod 8) since 8 | (17 − 1). Notice that 1 ≡ 17 (mod 8) as well, since 8 | (1 − 17). If x ≡ y (mod m), we say that x is congruent to y mod m.

Theorem. If a ≡ b (mod m) then ac ≡ bc (mod m).

Proof. Choose arbitrary values a, b, c and m such that

(1) a ≡ b (mod m)

By the definition of what fact (1) means,

(2) m | a − b

By the definition of what fact (2) means,

(3) m ≠ 0
(4) ∃k (a − b = km)

So we can choose a particular value x for k so that

(5) (a − b = xm)

Multiplying both sides of that equation by c yields

(7) (ac − bc = xmc)

Thinking of j = xc, we see that

(8) ∃j(ac − bc = jm)

But using Definition 1 again, but this time in the opposite direction, and remembering fact (3),

(9) m | ac − bc

Now using Definition 2 again, that means

(10) ac ≡ bc (mod m)

and we are done.

♦