Proving the Evenness of n^3: Simple Verification for Homework

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In summary, simple proof, also known as verification, is the process of confirming the validity of a statement or hypothesis using logical reasoning and evidence. It is important in science as it allows for the validation and replication of results. The steps involved in simple proof include formulating a hypothesis, designing an experiment, collecting and analyzing data, and drawing conclusions. The main difference between simple proof and complex proof is the level of complexity and rigor involved. Simple proof can be used in all scientific fields and is a fundamental aspect of the scientific method.
  • #1
NATURE.M
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Homework Statement



Let n be an integer. Prove that if n^3 is even, then n is even.

Homework Equations





The Attempt at a Solution



Suppose n is even. That is n=2m, for some mεZ.
Then, n^3=(2m)^3=8m^3=2(4m^3).
Since 4m^3 is an integer, n^3 will be even.

Now, i proved that if n is even, then n^3 is even. So would this be a valid proof in this context?
 
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  • #2
NATURE.M said:

Homework Statement



Let n be an integer. Prove that if n^3 is even, then n is even.

Homework Equations





The Attempt at a Solution



Suppose n is even. That is n=2m, for some mεZ.
Then, n^3=(2m)^3=8m^3=2(4m^3).
Since 4m^3 is an integer, n^3 will be even.

Now, i proved that if n is even, then n^3 is even. So would this be a valid proof in this context?

Looks fine.

Edit: I misread the problem statement as "if n is even then n3 is even." My mistake.
 
Last edited:
  • #3
Your proof is showing that if n is even then n^3 is even.

You want to show the converse that if n^3 is even then n is even.

It is semantics and may seem silly in this sort of proof but in more difficult proofs you have to be careful of what you are saying.

Edit: Take the factors of n^3 and see what that means to be even.


Assume n^3 is even and n is an integer. Well, n^3 = n* n* n. Since n^3 is even n*n*n is even which means that either n or n or n is divisible by two. Therefore, n is an integer divisible by two; n is even.
 
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  • #4
Jufro said:
Your proof is showing that if n is even then n^3 is even.

You want to show the converse that if n^3 is even then n is even.

It is semantics and may seem silly in this sort of proof but in more difficult proofs you have to be careful of what you are saying.

Assume n^3 is even and n is an integer. Well, n^3 = n* n* n. Since n^3 is even n*n*n is even which means that either n or n or n is divisible by two. Therefore, n is an integer divisible by two; n is even.

It's good to point out what he did wrong, but I presume you know you are not supposed to supply complete solutions.
 
  • #5
I just wanted to make sure that I wasn't missing something.

To comply better with PF rules I updated my post
 
  • #6
So many of these problems are made easier when you look at the contrapositive of the claim. Here it is just that if n is odd, then n^3 is odd.
 
  • #7
Jufro said:
Your proof is showing that if n is even then n^3 is even.

You want to show the converse that if n^3 is even then n is even.

It is semantics and may seem silly in this sort of proof but in more difficult proofs you have to be careful of what you are saying.

Edit: Take the factors of n^3 and see what that means to be even.


Assume n^3 is even and n is an integer. Well, n^3 = n* n* n. Since n^3 is even n*n*n is even which means that either n or n or n is divisible by two. Therefore, n is an integer divisible by two; n is even.
Yeah, that's what I was really unsure about, but thanks for clarifying.
 
  • #8
verty said:
So many of these problems are made easier when you look at the contrapositive of the claim. Here it is just that if n is odd, then n^3 is odd.

The solution to the actual problem uses the contrapositive method you described, where n is odd, then n^3 is odd. Though, I haven't studied contrapositive proofs yet, so I posted an alternative method above.
 
  • #9
What you proved was the converse, not the contrapositive.

If you're uncertain about these terms, maybe this will help.

Implication P => Q (if P then Q)
Converse: Q => P
Contrapositive: ~Q => ~P (if not Q, then not P)

There's another term for the converse of the contrapositive, but I forget what it is.

An implication and its contrapositive are always equivalent (i.e., they have exactly the same truth table values), but an implication and its converse are not equivalent.

Here are some examples.
Implication: If x = 2, then x2 = 4

Converse of the implication: If x2 = 4, then x = 2 (not necessarily true, as x could be -2)

Contrapositive of the implication: If x2 ≠ 4, then x ≠ 2
 
  • #10
Mark44 said:
What you proved was the converse, not the contrapositive.

If you're uncertain about these terms, maybe this will help.

Implication P => Q (if P then Q)
Converse: Q => P
Contrapositive: ~Q => ~P (if not Q, then not P)

There's another term for the converse of the contrapositive, but I forget what it is.

An implication and its contrapositive are always equivalent (i.e., they have exactly the same truth table values), but an implication and its converse are not equivalent.

Here are some examples.
Implication: If x = 2, then x2 = 4


Converse of the implication: If x2 = 4, then x = 2 (not necessarily true, as x could be -2)

Contrapositive of the implication: If x2 ≠ 4, then x ≠ 2

Thanks a lot Mark44. It makes sense to me now why the solutions manual proved the statement, "if n^3 is even, then n is even" by giving a proof of " if n is odd, then n^3 is odd", which is the contrapositive statement. This would be valid as the implication statement is equivalent to its contrapositive statement, as you pointed out.
 

What is simple proof?

Simple proof, also known as verification, is the process of confirming the validity of a statement or hypothesis through logical reasoning and evidence.

Why is simple proof important in science?

Simple proof is important in science because it allows researchers to validate their theories and findings, ensuring that they are accurate and reliable. It also allows for the replication of results, which is a key aspect of the scientific method.

What are the steps involved in simple proof?

The steps involved in simple proof include formulating a hypothesis, designing a controlled experiment, collecting and analyzing data, and drawing conclusions based on the evidence.

What is the difference between simple proof and complex proof?

The main difference between simple proof and complex proof is the level of complexity and rigor involved. Simple proof typically involves straightforward logical reasoning and evidence, while complex proof may require more advanced mathematical or scientific techniques and data analysis.

Can simple proof be used in all scientific fields?

Yes, simple proof can be used in all scientific fields. It is a fundamental aspect of the scientific method and is used to validate theories and findings in various disciplines such as biology, chemistry, physics, and more.

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