- #1

Keen94

- 41

- 1

## Homework Statement

*Given a series of mathematical statements, some of which are true and some of which are false. Prove those of which are true, and disprove those which are false.*

1. The sum of three odd integers is odd.

Text:

1. The sum of three odd integers is odd.

Text:

__Principles of Mathematics__by Allendoefer and Oakley.## Homework Equations

∀

_{x}(p

_{x}→q

_{x})↔P⊆Q

*Law of Detachment and Law of Substitution.*

## The Attempt at a Solution

I am mainly looking for feedback on my notation (formalism). What is the proper way of writing out a proper basic proof?

Statement: The sum of three odd integers is odd.

∀

_{x}(integers): If x is odd, then ∑

^{3}

_{i=1}x

_{i}is odd.

(1) Assume x

_{1}, x

_{2}, and x

_{3}is odd.

**[Hypothesis]**

(2) The integers a, b, and c exist such that x

_{1}=2a+1, x

_{2}=2b+1, and x

_{3}=2c+1.

**[Def**

^{n}of odd](3) ∑

^{3}

_{i=1}x

_{i}=(2a+1)+(2b+1)+(2c+1)

=2a+1+2b+1+2c+1

=2a+2b+2c+2+1

=2(a+b+c+1)+1

=2(m)+1

**[Let m=a+b+c+1]**

Therefore the sum of three odd integers is odd by definition.