Basic Proof Writing: Prove Sum of 3 Odd Ints is Odd

[Keen94]

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: 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=1x_i is odd.
(1) Assume x₁, x₂, and x₃ is odd. [Hypothesis]
(2) The integers a, b, and c exist such that x₁=2a+1, x₂=2b+1, and x₃=2c+1. [Defⁿ of odd]
(3) ∑³_i=1x_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.

 

[Mark44]

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: 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=1x_i is odd.

Not wrong, but writing $\sum_{i = 1}^3 x_i$ seems like overkill here. Just let x, y, and z be the odd integers. Their sum is x + y + z.

(1) Assume x₁, x₂, and x₃ is odd. [Hypothesis]
(2) The integers a, b, and c exist such that x₁=2a+1, x₂=2b+1, and x₃=2c+1. [Defⁿ of odd]
(3) ∑³_i=1x_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.

The gist of your proof is fine, but you're using something that is in my opinion unnecessary (the summation and subscripted variables).

 

[Keen94]

Thank for replying Mark44 and taking the time to help me. Would the following adjustment make the proof less overkill?

Statement: The sum of three odd integers is odd.
∀_x∀_y∀_z(integers): If x, y, and z are odd, then x+y+z is odd.
(1) Assume x, y, and z are odd [Hypothesis]
(2) The integers a, b, and c exist such that x=2a+1, y=2b+1, and z=2c+1. [Defⁿ of odd]
(3) x+y+z =(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 [Defⁿ of odd]
Therefore x+y+z is odd by definition of odd.

 

[Mark44]

That looks good. The only think I would add is that it wouldn't hurt (IMO) to reduce the formalism a bit more in your statement of the problem. Instead of saying "∀x∀y∀z(integers): If x, y, and z are odd, then x+y+z is odd," you could say the same thing as "For any integers x, y, and z, x + y + z is odd." All the rest looks fine.

 

[Keen94]

That looks good. The only think I would add is that it wouldn't hurt (IMO) to reduce the formalism a bit more in your statement of the problem. Instead of saying "∀x∀y∀z(integers): If x, y, and z are odd, then x+y+z is odd," you could say the same thing as "For any integers x, y, and z, x + y + z is odd." All the rest looks fine.

Thanks for the feed back Mark44. Going forward I will have more questions about this topic. Should I keep creating new threads, as in a new thread per question or just keep everything under one thread? Thanks.

 

[Mark44]

Thanks for the feed back Mark44. Going forward I will have more questions about this topic. Should I keep creating new threads, as in a new thread per question or just keep everything under one thread? Thanks.

New question -- new thread.

 

[Keen94]

Gotcha, see you there haha.