B.1 What is a definition?
Ambiguity is the drawback of natural languages. How would you define, say, the concept of “hot”? Upon several attempts, you would soon discover that no two people have the same definition.
In mathematics, there is no room for ambiguity. Every object and every property must be precisely defined. It’s best to look at a good example instead of philosophizing about it.
Definition 103. (Divisors)
Let b ∈ℤ be an integer. We say that a ∈ℤ is a divisor of b if there exists an integer k ∈ℤ such that b = ka.
The property “a is a divisor of b” is denoted by a∣b.
For example, 2∣10 and 5∣10, but 7 ∤ 10. (Crossed symbols mean the negation of the said property.)
In terms of our formal language, the definition of “a is a divisor of b” can be written as
Don’t let the a∣...
