For the webpage title: Proving the Oddness of 5x-3 for Even x Using Direct Proof

[TylerH]

Homework Statement

Prove by direct proof that 5x-3, where x is an integer, is odd for all even x.

Homework Equations

N/A

The Attempt at a Solution

Let x=2a, where a is an integer. It follows that

5x-3=10a-2-1=2(5a-1)-1.​

Although it is obvious, I don't know how to logically prove 2(5a-1)-1 is odd. How do I do that?

 

[micromass]

Hi TylerH!

How did you define "odd" in this case?? There are a lot of possible definitions, and some will make this easy to prove, some hard.

 

[TylerH]

The book I'm working in gives the definition that numbers are odd iff they are of the form 2k+1, where k is an integer. Equivalence relations come to mind, but they're a few chapters past this, so they shouldn't be necessary.

 

[HallsofIvy]

Well, 2(5a-1)- 1= 2(5a-1)- 2+ 1= 2(5a- 3)+ 1 and 5a- 3 is an integer.

(Of course, it is also true that any integer of the form 2a- 1 is odd.)

Thanks to magicarpet215 for pointint out my mistake.

 

[micromass]

Ah, that's one of the easy definitions:

Try to write 2(5a-1)-1 in the form 2n+1 for a certain n. So, you need to find an n. How to do this? Well, put 2(5a-1)-1=2n+1 (we want this right?) and work out what n is...

 

[magicarpet512]

(Of course, it is also true that any integer of the form 2a- 1 is even.)

Shouldn't this say 2a - 1 is odd?

 

[TylerH]

OHH! That was SO simple. I should've thought of that.

Thanks!

 

[gb7nash]

Shouldn't this say 2a - 1 is odd?

Yes. Any even number (in this case 2a) plus or minus any odd number (in this case -1) is odd.

 

[HallsofIvy]

Shouldn't this say 2a - 1 is odd?

Yes. I will now go back and edit my answer so I can pretend I never said that!

 

[TylerH]

Let x=2a, where a is an integer. It follows that

5x-3=10a-4+1=2(5a-2)+1. [insert end of proof box here]​

Would this be an acceptable proof for the assignment? Do I need to show that 2(5a-2)+1 is of the form 2k+1 or is it commonly allowed to be assumed the audience is capable? This is one of my first proofs; I'm preparing for the class, but it has yet to start. Otherwise, I would ask my professor.

 

[magicarpet512]

I would say:

Since a is an integer then (5a - 2) is also an integer, namely k.
Therefore, (whatever number) is odd since its of the form 2k + 1.

 

[micromass]

No, this is good. BUT something you really need to learn is how to write proofs. You can't just have a string of equations, you need some words to glue the equations together. A common mistake with beginners is that they don't do this.

Your proof is correct of course, but try to put some work in presenting it. How I would write your proof is:

Our goal is to prove that 5x-3 is odd for even integers x. So, assume that x=2a. If we substitute that into 5x-3, we get

$$5x-3=10a-3=10a-4+1=2(5a-2)+1$$

So, we have shown that 5x-3 has the form 2n+1, where n=5a-2. Hence, 5x-3 is odd.
​

As you see, this proof is quite longer. But it much more pleasant to read! In fact, if you search research papers, you will often see long strings of text with occasionally some symbols and some equations. This is the way mathematical texts should be written...

 

[jhae2.718]

Don't forget the Q.E.D. at the end!

If you prefer the black square thing :(, here's one: ■

 

[micromass]

Don't forget the Q.E.D. at the end!

I actually hate Q.E.D. at the end of proofs I usually write a box: $$\Box$$. Although I've seen people using smileys at the end of proofs, which is irritating if you don't get the proof: you think the auther is laughing at you...

 

[magicarpet512]

Don't forget the Q.E.D. at the end!

If you prefer the black square thing :(, here's one: ■

Are these always necessary at the end of a proof?
I hardly ever use these on homeworks or exams, unless I have more than one proof on one page.
But of course i see the importance of using these in a more formal setting like writing a paper.

 

[micromass]

Are these always necessary at the end of a proof?
I hardly ever use these on homeworks or exams, unless I have more than one proof on one page.
But of course i see the importance of using these in a more formal setting like writing a paper.

It's not necessary of course. But it makes it easier to read. If I were you, I would always include them. But the proof isn't wrong if you don't include it, of course

 

[TylerH]

No, this is good. BUT something you really need to learn is how to write proofs. You can't just have a string of equations, you need some words to glue the equations together. A common mistake with beginners is that they don't do this.

Your proof is correct of course, but try to put some work in presenting it. How I would write your proof is:

Our goal is to prove that 5x-3 is odd for even integers x. So, assume that x=2a. If we substitute that into 5x-3, we get

$$5x-3=10a-3=10a-4+1=2(5a-2)+1$$

So, we have shown that 5x-3 has the form 2n+1, where n=5a-2. Hence, 5x-3 is odd.
​

As you see, this proof is quite longer. But it much more pleasant to read! In fact, if you search research papers, you will often see long strings of text with occasionally some symbols and some equations. This is the way mathematical texts should be written...

I get the "proofs are more than just equations and formulas" thing. It's just that, after being indoctrinated for so long that the right way to solve a problem has a template, and all you have to do is fill in that template, I find it hard to know what strays too far from that template. Hence the use of "It follows that," which I'm sure you recognize as one of the most used phrases in proofs.

Also, I remember from high school geometry, in which we wrote simple proofs, we were required to state the axioms or theorems on which the proof relies. Is this practice also expected in higher math?

For example, would this be better than the previous:

Assume x=2a, by substitution property of equality, it follows that

5x-3=10a-4+1=2(5a-2)+1.​

Given that the integers are closed under multiplication and addition and that integers are odd iff they can be written in the form 2k+1, where k is an integer, we can see that 5a-2 is also an integer, which implies that 2(5a-2)+1 is of the form 2k+1, and is therefore odd. Since, by assuming x is even, we may rewrite 5x-3 in the form of an odd integer, we have shown 5x-3 to be odd for all even x.

 

[magicarpet512]

TylerH:
In your intro to proofs class you will quite often find yourself being pretty wordy in your proofs by using phrases such as " by substitution property of equality, it follows that." This is perfectly fine, and I would encourage you to be wordy like this, it will help you see what needs to be "filled into the gaps" of the proofs that you will encounter in your more advanced classes. You definitely need to learn proofs by filling in all the details that you can.

Once you get into your more advanced math courses and you are getting more comfortable writing proofs, you will eventually find yourself venturing away from proofs that are too wordy. You will probably find yourself beginning to imitate how your professors and books prove theorems to you by using phrases like "clearly such and such is true..." and moving on to more interesting parts of your proofs.

But for now, just focus on all of the details that go into a proof. I would say go ahead and state definitions in your proofs, its a good way to learn the definitions. Dont worry too much right now about being too wordy; over time you will learn what to say and not to say. It takes lots and lots of practice to tweak your proofs into what people want to read, and they probably won't get that way your first semester of proof writing.

 

[icystrike]

Lol. I prefer qed simply because it means quite easily done to me:)

 

[magicarpet512]

Assume x=2a, by substitution property of equality, it follows that...

I should of thought about this earlier, but to some people might not think that "by substitution property of equality" makes sense.
The grader might count off and say "by substitution and equality of what?"

I'm sure you will come across some kind of Uniqueness & Existence theorem one of these days. I have heard several professors complain that their students may write "by uniqueness and existence, such and such is true;" where the professors would have rather seen "by uniqueness and existence OF WHAT, such and such is true."

So if you are going to use phrases like "by substitution property of equality," then go ahead and include the "of what" part in your proof.

As with your proof above, I think it would be understandable to simply say
"... by substitution of x, it ffollows that..."

But like I said earlier, it takes lots of practice, and consulting with professors and other students, etc, for you to write really good proofs; and you're off to a good start!