Proving Equivalence Relations on the Cartesian Coordinate Plane

[playboy]

I'm doing this problem in the book - their are 2 of this kind and they have no answers in the back.. so i thought ill post one.

Let S be the Cartesian coodinate plane R x R and define a relation R on S by (a,b)R(c,d) iff a+d=b+c. Verify that R is an equivalence relation and describe the equivalence class E(7,3)

...So i want to show that the 3 properties of an equivalence reltion holds: (reflective propery, symmetric propery and transitive propery.)

1. reflective propery

(a,b)=(a,b) and (c,d)=(c,d)...(not hard)

2. symmetric propery

Let (a,b) be in A and (c,d) be in B and if A R B, then (a,b) = (c,d)
so, (c,d) = (a,b) and thus, B R A.

How does that sound?

3. transitive propery

i have no idea how to start this

i mean, the property states "if xRy and yRz, then xRz"

so will this work:?

if (a,b)=(c,d) and (c,d)=(e,f), then (a,b)=(e,f)

but then i feel like i making things up when i put in (e,f)?

 

[Hurkyl]

If you're trying to prove "=" is an equivalence relation on ordered pairs, then you've correctly stated what needs to be proved, at least for (1) and (3).

Let (a,b) be in A and (c,d) be in B and if A R B

Where did "A" and "B" come from?


But this is all moot, since you're trying to prove "R" is an equivalence relation.

 

[playboy]

I was trying to get fancy with the A and B.
My prof did that in class today (but he was talking about similar triangles and stated "Let A be so and so, and Let be be so and so.. etc..)

so i guess 2 would be something like this:

(a,b) = (c,d) and (c,d) = (a,b)

Just 2 more things.

1) where does a+b = c+d come in?
2)What does it mean "and describe the equivalence class E(7,3)"
I suppose E(7,3) assumes a=7 and b = 3. So won't that be something like (a,b)=(c,d)=(7,3)

 

[Hurkyl]

Again, that's what the symmetric relation would look like, for "=".


Incidentally, what you could say is something akin to:

Let A, B be in S.
We may write A = (a, b) and B = (c, d) for some a, b, c, d.
(more stuff)

1) where does a+b = c+d come in?

It's the definition of (a,b)R(c,d), as you stated in your original post.

I suppose E(7,3) assumes a=7 and b = 3.

Where did a and b come from! You're falling into a trap -- let me demonstrate it with a common mistake people make:


What you just said is like "defining" the function f via:

f = x²

this is nonsense: where did x come from? Now, if you introduced x as a dummy variable, you could define f pointwise via:

f(x) = x²

which does make sense, and means exactly the same thing as

f(a) = a²

or, as one professor I knew would sometimes write to emphasize this point:

$$f(\spadesuit) = \spadesuit^2$$

 

[matt grime]

None of your 'proofs' that R is an equivalence relation mention R. You might want to think why that is not a good thing. If I wished to show that (a,b)R(a,b) ie it were reflexive then I might want to consider what R actually is.

 

[playboy]

Matt Grimme:

the question said "Verify that R is an equivalence relation" so R would be "="

 

[Hurkyl]

Why would "R" be "="?

"=" is an equivalence relation. Not all equivalence relations are "=".

 

[playboy]

okay i think I am on something:

Q) Let S be the Cartesian coodinate plane R x R and define a relation R on S by (a,b)R(c,d) iff a+d=b+c. Verify that R is an equivalence relation and describe the equivalence class E(7,3)

check: reflective propery, symmetric propery and transitive propery.

reflective propery:

is (a,b)R(a,b)?
(a,b)R(a,b) iff a+b=b+a

since a+b = b+a, the relationship is reflective.

symmetric propery

is (a,b)R(c,d) the same as (c,d)R(a,b)

(a,b)R(c,d) iff a+d=b+c
and (c,d)R(a,b) iff c+b=d+a
thus we have a+d=b+c and c+b=d+a which are the same, thus the symmetric property holds

transitive propery:

if (a,b)R(c,d) and (c,d)R(e,f) the (a,b)R(e,f)
so, if a+d=b+c and c+f=d+e, then a+f = b+e (which can easily be shown with some arithmatic)

thus, the tranistive property holds.

How does this sound?

 

[Hurkyl]

Looks good!

 

[playboy]

oh boo-yeah, i feel good now!
Thank you!

but i still don't get what "discrie the equivelence class E(7,3)" means?

 

[Hurkyl]

I suppose (based on the explanatory text you gave) they're using the "E" to mean "the equivalence class of". So, EX means the equivalence class of X!

 

[playboy]

the question says "verify that R is an equivalence relation and describe the equivalence class E(7,3)"

So, based on (a,b)R(c,d) iff a+d = b+c ... i came up with this:

(5,2)R(2,1) = (7,3) since 5+2=7 and 2+1=3

but that's just one example... i have no idea

 

[daveb]

What would happen if you wrote (7,3)R(a,b)?

 

[matt grime]

the question says "verify that R is an equivalence relation and describe the equivalence class E(7,3)"
So, based on (a,b)R(c,d) iff a+d = b+c ... i came up with this:
(5,2)R(2,1) = (7,3) since 5+2=7 and 2+1=3
but that's just one example... i have no idea

Where did that come from? The equivalence class of something is the set of all elements related to it.

 

[playboy]

The equivlence class is something like this:
E(7,3) = { (c,d) in S: (c,d)R(7,3) }

from what daveb wrote, What would happen if you wrote (7,3)R(c,d)?

you would get
7 + d = 3+ c
4 = c - d

so we want all the points (c,d) that satisfy 4 = c - d.

such possibilities are (4,0) (0,-4) (6, 2) (8,4) etc... and we notice from observation (and we could so some aritimatic to show) that the points all lie on the line y = x - 4.

Thus, "discribe the equivalence class E(7,3)" is the line y = x -4

How does that sound?

 

[dglee]

testing testing

 

[matt grime]

The equivlence class is something like this:
E(7,3) = { (c,d) in S: (c,d)R(7,3) }
from what daveb wrote, What would happen if you wrote (7,3)R(c,d)?[\quote]


it's an equivalence relation, it's symmetric, what do you think would happen?

 

[HallsofIvy]

The equivlence class is something like this:
E(7,3) = { (c,d) in S: (c,d)R(7,3) }
from what daveb wrote, What would happen if you wrote (7,3)R(c,d)?
you would get
7 + d = 3+ c
4 = c - d
so we want all the points (c,d) that satisfy 4 = c - d.
such possibilities are (4,0) (0,-4) (6, 2) (8,4) etc... and we notice from observation (and we could so some aritimatic to show) that the points all lie on the line y = x - 4.
Thus, "discribe the equivalence class E(7,3)" is the line y = x -4
How does that sound?

NOW you've got it!

 

[playboy]

now I am happy, thanks everyone :)