Convention notes notes notes 2 ez ozatla

Notes

More Definitions
Definition
Suppose a and b are integers. We say a divides b , written
a | b
if b = ac for some c 2 Z . In this case we also say that a is a
divisor of b , and b is a multiple of a .
Example: 10 = 2 · 5
I
2 | 10, 5 | 10
I
2 is a divisor of 10
I
10 is a multiple of 2
I
Is 0 a divisor of 10? No. There is no a 2 Z with 0 a = 10.
We write 0 6 | 10
I
Is - 1 a divisor of 10?
Notes
3

Direct Proof of Conditional Statements
How to prove P ) Q
Proposition: If P , then Q .
Proof : Suppose P .
.
.
.
Therefore Q .
Every step should be completely justified.
P Q P ) Q
T T T
T F F
F T T
F F T
Direct Proof
Prove the following:
If 4 | x , then
x
2
is even.
Proposition: If 4 | x , then
x
2
is even.
Proof : Suppose 4 | x .
Then by the definition of “divides,” there exists some
b 2 Z such that 4 b = x .
Then
x
2
=
4 b
2
= 2 b
Therefore,
x
2
= 2 ab for some ab 2 Z .
Therefore,
x
2
is even by the definition of “even”. Then
there exists some a 2 Z such that 4 a = x . Then
x
2
= 2 a , so
x
2
is even.
Direct Proof
Prove the following:
If x is odd, then x 2 + 1 is even.
Proposition: If x is odd, then x
2
+ 1 is even.
Proof : Suppose x is odd.
Then x = 2 a + 1 for some integer a .
Then x
2
+1 = (2 a +1)
2
+1 = 4 a
2
+4 a +1+1 = 2(2 a
2
+2 a +1).