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

• Keen94
In summary, the conversation discusses the proof of the statement that the sum of three odd integers is odd. The proof involves using the definition of odd integers and basic algebraic manipulation. The conversation also includes a discussion on the proper way to write out a basic proof using formalism and notation.
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(px→qx)↔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 ∑3i=1xi 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) ∑3i=1xi =(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.

Keen94 said:

## 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(px→qx)↔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 ∑3i=1xi 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.

Keen94 said:
(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) ∑3i=1xi =(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).

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.
xyz(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.

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.

Mark44 said:
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.

Keen94 said:
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.

Gotcha, see you there haha.

## What is the definition of an odd integer?

An odd integer is any integer that is not divisible by 2. This means that when divided by 2, the result will always have a remainder of 1.

## Can you prove that the sum of three odd integers is odd?

Yes, we can prove this using the definition of an odd integer and basic algebraic properties.

## What are the steps to prove the sum of three odd integers is odd?

The steps to prove this are as follows:

1. Assume that a, b, and c are three odd integers.
2. Express the sum of a, b, and c as a single expression, such as a + b + c.
3. Use the definition of an odd integer to show that each individual term is odd.
4. Use basic algebraic properties to simplify the expression and show that the result is odd.
5. Therefore, the sum of three odd integers is odd.

## Why is it important to prove the sum of three odd integers is odd?

Proving this statement is important because it helps us understand the fundamental properties of odd integers and their behavior when combined with other odd integers. This can also serve as a building block for more complex mathematical proofs.

## Can the same proof be applied to the sum of any number of odd integers?

Yes, the same proof can be applied to the sum of any number of odd integers. This is because the definition of an odd integer and basic algebraic properties hold for any number of terms.

• Precalculus Mathematics Homework Help
Replies
4
Views
2K
• Precalculus Mathematics Homework Help
Replies
11
Views
4K
• General Math
Replies
3
Views
888
• Precalculus Mathematics Homework Help
Replies
1
Views
3K
• Precalculus Mathematics Homework Help
Replies
2
Views
2K
• Precalculus Mathematics Homework Help
Replies
11
Views
2K
• Precalculus Mathematics Homework Help
Replies
14
Views
4K
• Calculus and Beyond Homework Help
Replies
3
Views
748
• Precalculus Mathematics Homework Help
Replies
7
Views
1K
• Precalculus Mathematics Homework Help
Replies
9
Views
3K