Proving Equivalence: Cos^2(x) + Sin^2(y) = 1

[naaa00]

Homework Statement

The question is to show that Cos^2(x) + Sin^2(y) = 1 is an equivalence relation.

The Attempt at a Solution

I know that there are three conditions which the equation must satisfy. (reflexivity, symetry, transitivity)

For reflexivity I tried: Cos^2(x) - Cos^2(x) = Sin^2(y) - Sin^2(y) = 0.

For symmetry: ( I don't fully understand this condition) I think that maybe Sin^2(y) + Cos^2(x) = 1 will do. Why? I'm not sure. Any explanation will be very apreciated.

Transitivity: Cos^2(x) + Sin^2(y) = 1, Sin^2(y) + Cos^2(z) = 1, Cos^2(z) + Sin^2(h) = 1.

If I add all o them I'll get Cos^2(x) + Sin^2(y) + Sin^2(y) + Cos^2(z) + Cos^2(z) + Sin^2(h) = 1 + 1 + 1

I was tempted to say that Sin^2(y) = Sin^2(z) = Sin^2(h), and that cos^2(x) = cos^2(z) (my thought was: "since they are supposed to be equivalent...") and rewrite it as:

3Cos^2(x) + 3Sin^2(h) = 3 (and then I divide it by 3)

Cos^2(x) + Sin^2(h) = 1.

I think my methods are wrong.

Any help will be very appreciated.

 

[SteveL27]

It will help to write the problem more precisely, then say exactly what you're trying to show in each step.

The problem should be stated like this:

Define x ~ y if cos^2(x) + sin^2(y) = 1. Show that ~ is an equivalence relation.

Now, write down exactly what is meant by reflexivity, transitivity, and transitivity in terms of ~. This will get you on the right track regarding what you need to prove. Start with reflexivity.

 

[naaa00]

Hello there!

well, I have read many times the definitions of each, but still don't get it.

a) x ~ x, (reflexivity)

Cos^2(x) = Cos^2(x) (?)

b) if x ~ y and y ~ z then x ~ z. (transitivity)

Cos^2x + Sin^2y, (?)

c) if x ~ y then y ~ x (symmetry)

Cos^2x = Cos^2y, Sin^2y = Cos^2x (?)

Why you mentioned asscociativity?

 

[naaa00]

a) x ~ x, (reflexivity)

Cos^2(x) + Sin^2(x)= 1, Cos^2x = 1 - (1 - Cos^2x), Cos^2x = Cos^2x. (?)

 

[SteveL27]

a) x ~ x, (reflexivity)

Cos^2(x) + Sin^2(x)= 1, Cos^2x = 1 - (1 - Cos^2x), Cos^2x = Cos^2x. (?)

Yes, one of the above is correct but why?

What is the exact definition of ~? What would the exact definition of x ~ x be? Is it true?

 

[naaa00]

a) x ~ x, (reflexivity)

Cos^2x + Sin^2x = 1

b) if x ~ y and y ~ z then x ~ z. (transitivity)

Cos^2x + Sin^2y = 1, Cos^2y + Sin^2z = 1,

Cos^2x + Sin^2y + Cos^2y + Sin^2z = 2,

(Cos^2x + Sin^2z) + (Cos^2y + Sin^2y) = 2

c) if x ~ y then y ~ x (symmetry)

Cos^2x + Sin^2y = 1, Cos^2y + Sin^2x = 1,

Right?

 

[naaa00]

Yes, one of the above is correct but why?

What is the exact definition of ~? What would the exact definition of x ~ x be? Is it true?

eh, I guess because there is an x, for which the definition of ~ (in this case) holds true, namely Cos^2x + Sin^2x =1, right?

 

[SteveL27]

eh, I guess because there is an x, for which the definition holds true, namely Cos^2x + Sin^2x =1, right?

Do you understand that reflexivity says that x ~ x must be true for ALL x in order for ~ to be an equivalence relation?

And in this specific case, is x ~ x always true for any x?

 

[naaa00]

I would say yes...

 

[naaa00]

So, I guess now my answer is correct, or? I'll try another example, and see if I understand this.

x ~ y if e^(x^2)= e^(y^2). Show that ~ is an equivalence relation.a) x ~ x, (reflexivity)

e^(x^2)= e^(x^2), for all x.

b) if x ~ y and y ~ z then x ~ z. (transitivity)

e^(x^2)= e^(y^2),
e^(y^2)= e^(z^2),

e^(x^2)+ e^(y^2)= e^(y^2) + e^(z^2), (I cancel the e^(y^2))

e^(x^2)= e^(z^2),

c) if x ~ y then y ~ x (symmetry)

e^(x^2)= e^(y^2).

Is this correct?

 

[SteveL27]

I would say yes...

Did your teacher explain to you that you have to say WHY something is true?

Now you have to do transitivity and symmetry, right? Before you go on to the next one.

But you haven't completed reflexivity yet.

We have to show that for all x, x ~ x. That means we have to show that for all x, cos^2(x) + sin^2(x) = 1. And we know this is true because ...

 

[naaa00]

I know that it comes from the Pythagoras formula x^2 + y^2 = r^2. I substitute, Cos(t) = x/r, Sin(t)=y/r. And,

(x/r)^2+(y/r)^2 = (x^2+y^2)/r^2 = r^2/r^2 = 1.

Cos^2t + Sin^2t = 1

So, for reflexivity it will work for every t.

 

[SteveL27]

So, I guess now my answer is correct, or? I'll try another example, and see if I understand this.

x ~ y if e^(x^2)= e^(y^2). Show that ~ is an equivalence relation.


a) x ~ x, (reflexivity)

e^(x^2)= e^(x^2), for all x.

b) if x ~ y and y ~ z then x ~ z. (transitivity)

e^(x^2)= e^(y^2),
e^(y^2)= e^(z^2),

e^(x^2)+ e^(y^2)= e^(y^2) + e^(z^2), (I cancel the e^(y^2))

e^(x^2)= e^(z^2),

c) if x ~ y then y ~ x (symmetry)

e^(x^2)= e^(y^2).

Is this correct?

Yes I think this one's good.

 

[SteveL27]

I know that it comes from the Pythagoras formula x^2 + y^2 = r^2. I substitute, Cos(t) = x/r, Sin(t)=y/r. And,

(x/r)^2+(y/r)^2 = (x^2+y^2)/r^2 = r^2/r^2 = 1.

Cos^2t + Sin^2t = 1

So, for reflexivity it will work for every t.

Yes, in fact that's a little overkill. It's sufficient to note that cos^2(x) + sin^1(x) = 1 for every x. I think you've got the idea now.

 

[naaa00]

c) if x ~ y then y ~ x (symmetry)

e^(x^2)= e^(y^2).

e^(y^2)= e^(x^2).

 

[naaa00]

but does transitivity and symmetry are correct on the first example?

 

[SteveL27]

but does transitivity and symmetry are correct on the first example?

You wrote:

"I think that maybe Sin^2(y) + Cos^2(x) = 1 will do. Why? I'm not sure. Any explanation will be very apreciated."

That's not right. You should write down EXACTLY what it is you're trying to show, starting with defining what it means for ~ to be symmetric.

 

[naaa00]

Hello there! I did indeed rewrote it. It's post #6. I will rewrite it here:

a) x ~ x, (reflexivity)

Cos^2x + Sin^2x = 1, for all x.

b) if x ~ y and y ~ z then x ~ z. (transitivity)

Cos^2x + Sin^2y = 1, Cos^2y + Sin^2z = 1,

Cos^2x + Sin^2y + Cos^2y + Sin^2z = 2,

(Cos^2x + Sin^2z) + (Cos^2y + Sin^2y) = 2

c) if x ~ y then y ~ x (symmetry)

Cos^2x + Sin^2y = 1, Cos^2y + Sin^2x = 1,

Is this fine?

 

[SteveL27]

c) if x ~ y then y ~ x (symmetry)

Cos^2x + Sin^2y = 1, Cos^2y + Sin^2x = 1,

Is this fine?

Well you wrote down what you have to prove, but you didn't prove it. I didn't look at the transitivity proof.

What kind of class is this for?

 

[naaa00]

Hello there!

Let's see:

c) if x ~ y then y ~ x (symmetry)

Cos^2x + Sin^2y = 1, Cos^2y + Sin^2x = 1,

(1 - Sin^2x) + (1 - Cos^2y) = 1

2 - Sin^2x - Cos^2y = 1, - Sin^2x - Cos^2y = -1 (I multiply by -1)

Sin^2x + Cos^2y = 1,

Is this fine?

What kind of class? What do you mean? Course? I'm learning this because I want to.

 

[SteveL27]

What kind of class? What do you mean? Course? I'm learning this because I want to.

The reason I asked is that you usually learn about equivalence relations in set theory or abstract algebra. In which case you need to work on writing formal proofs.

Or, it could be a class in "how to write proofs." In which case you need to go see your teacher, because you are not grasping the material.

That's why I was curious to know what level math we're dealing with. I was not sure how much criticism to apply and in what fashion.

Basically you need to study or learn how to write a formal proof. A lot of people have trouble with that topic. I don't know how to advise you on self-learning that topic, other than looking at all the proofs in your book.

When you're learning about equivalence relationships, you typically are learning to write formal proofs at the same time. That's because equivalence relationships are one of the earliest bits of abstract math people are exposed to. So along with writing down the right equation, you also have to present your work in a logical format that you're having trouble with.

I was just casting about for more information so that I can respond appropriately in a manner that's helpful without being unduly critical. Anyway that's my two cents. Perhaps someone else can jump in and help me here :-)

 

[naaa00]

I see.

I got a book written by Polya, "how to solve it", but I haven't started reading it. And also a book of discrete mathematics. In this book, there is a chapter dedicated to this matter, but, again, I haven't started reading it...

But I will read both for sure, I just want to be done with some stuff before.

By the way, you didn't tell me if my answer for symmetry and transitivity are fine!

 

[SteveL27]

I see.

I got a book written by Polya, "how to solve it", but I haven't started reading it. And also a book of discrete mathematics. In this book, there is a chapter dedicated to this matter, but, again, I haven't started reading it...

But I will read both for sure, I just want to be done with some stuff before.

By the way, you didn't tell me if my answer for symmetry and transitivity are fine!

The symmetry one is essentially correct. If it were me I would want to see a more logically organized exposition, something like this:

Symmetry. We need to show that if x ~ y then y ~ x.

So, suppose x ~ y for two real numbers x and y. Then by definition

cos^2(x) + sin^2(y) =1 [by the way your leaving out those parens is bad form]

Using the trig relation cos^2(x) + sin^2(x) = 1, we can rewrite the above equation as

(1 - sin^2(x)) + (1 - cos^2(y)) = 1

=> 2 -(sin^2(x) + cos^2(y)) = 1

=> sin^2(x) + cos^2(y) = 1

=> cos^2(y) + sin^2(x) = 1

=> y ~ x, which is what we were required to prove.

 

[naaa00]

I see. So:

Transitivity. We need to show that if x ~ y and y ~ z, then x ~ z.

Suppose x ~ y and y ~ z for three real numbers x, y, and z. Then by definition

cos^2(x) + sin^2(y) = 1 and cos^2(y) + sin^2(z) = 1.

We can add both equations

cos^2(x) + sin^2(y) + cos^2(y) + sin^2(z) = 2,

and rewrite it as

[cos^2(x) + sin^2(z)] + [cos^2(y) + sin^2(y)] = 2.

Using the trig relation cos^2(y) + sin^2(y) =1, we find that

[cos^2(x) + sin^2(z)] + 1 = 2 or cos^2(x) + sin^2(z) = 1.

Thus x ~ z, which is what we were required to prove.

Better?

 

[SteveL27]

I see. So:

Transitivity. We need to show that if x ~ y and y ~ z, then x ~ z.

Suppose x ~ y and y ~ z for three real numbers x, y, and z. Then by definition

cos^2(x) + sin^2(y) = 1 and cos^2(y) + sin^2(z) = 1.

We can add both equations

cos^2(x) + sin^2(y) + cos^2(y) + sin^2(z) = 2,

and rewrite it as

[cos^2(x) + sin^2(z)] + [cos^2(y) + sin^2(y)] = 2.

Using the trig relation cos^2(y) + sin^2(y) =1, we find that

[cos^2(x) + sin^2(z)] + 1 = 2 or cos^2(x) + sin^2(z) = 1.

Thus x ~ z, which is what we were required to prove.

Better?

Yes! Good job.

 

[naaa00]

Hello! Yes! Finally! Thank you very much Steve!

 

[SteveL27]

You're very welcome :-)