# Interactive Real Analysis

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## 1.1. Notation and Set Theory

### Examples 1.1.5(a):

Prove that when two even integers are multiplied, the result is an even integer, and when two odd integers are multiplied, the result is an odd integer.
To prove this we first need to know what exactly an even and odd integer is:
• an integer x is even if x = 2n for some integer n
• an integer x is odd if x = 2n + 1 for some integer n
Now that we have a precise definition, the actual proof is easy: Take x and y two even numbers. Then
• x = 2n for some integer n
• y = 2m for some integer m
Multiplying these numbers together we get
• xy = (2n)(2m) = 4 nm = 2 (2nm) = 2 k
where k = 2nm. Hence, xy is again even.

If x and y are two odd numbers, then

• x = 2n + 1 for some integer n
• y = 2m + 1 for some integer m
Multiplying these numbers together we get
• xy = (2n+1)(2m+1) = 4nm + 2(n + m) + 1 = 2 (2nm + n + m) + 1 = 2k + 1
where k = 2nm + n + m. Hence, xy is again odd.
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