Rather we have started to get acquainted with the

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IMPORTAglyph1197T In a proof, we are given some information, and logic is employed to effect deduction,
that is, to make new conclusions from the given information.
In our example, we see that given two facts: Definition 1.1 and 8=2∙4 , it follows “logically” that 8 is
even. So there are two essential ingredients in a proof: inputted information and logic .
IMPORTAglyph1197T Definitions represents a major input in a proof. There is more than one way to define
something, and our proof depends on the definitions we use .
Example 1.3 We give an alternative definition of even integers, and a corresponding proof that 8 is even.
Definition 1.1a An integer ݔ is even if ݔ/2 is an integer.
Proof Since 8/2=4 , which is an integer, we have 8 is even by Definition 1.1a.
Example 1.4 Similarly, we can disprove that 7 is even (or equivalently, to prove that 7 is not even).
Proof 1 Since 7 is between 6=2ሺ3ሻ and 8=2ሺ4ሻ , and there is no integer between 2 and 3 , we know
that there is no integer ݊ such that 2݊=7 . So, by Definition 1.1, 7 is not even.
Proof 2 Since 7/2=3.5 , which is not an integer, 7 is not even by Definition 1.1a.
Example 1.5 How can we define an odd integer?
Definition 1.2 An integer ݔ is odd if ݔ=2݊+1 for some integer ݊ .
Definition 1.2a An integer ݔ is odd if ݔ is not even.
IMPOTAglyph1197T Definition 1.2a is legitimate because we have already defined an even integer.
Example 1.6 Prove that 7 is an odd number.
Proof 1 Since 7=2ሺ3ሻ+1 and 3 is an integer, by Definition 1.2, 7 is an odd number.
Proof 2 Since we have proved that 7 is not even, by Definition 1.2a, 7 is odd.
In Proof 2, we use a previously established fact, namely, 7 is not even (Example 1.4).
IMPOTAglyph1197T Apart from using definitions, we often use proven theorems or results in proofs.
We move beyond “even” and “odd”, and define another fundamental mathematical concept: divisibility .
Definition 1.3 An integer ܽ divides an integer ܾ (denoted ܽ|ܾ ) if ܾ=ܽ݊ for some integer ݊ .
Definition 1.4 An integer ܽ is a divisor ( factor ) of an integer ܾ if ܽ divides ܾ .
Definition 1.5 An integer ܾ is a multiple of an integer ܽ if ܽ is a divisor of ܾ .
IMPOTAglyph1197T
• “ ܽ divides ܾ ” is equivalent to “ ܽ is a divisor of ܾ ”, and “ ܾ is a multiple of ܽ ”.
• We introduce a notation : ܽ|ܾ . Notations are extremely important in mathematics! We use ܽ∤ܾ
to denote “ ܽ does not divide ܾ ”, just as ≠ denotes inequality. In general, “slash” means “not” .
Example 1.7 Suppose we have the following alternative definition of divisibility.
Definition 1.3a An integer ܽ divides an integer ܾ (denoted ܽ|ܾ ) if ܾ/ܽ is an integer.