Proving Equivalence Relations on the Cartesian Coordinate Plane

In summary, the conversation was about verifying that a relation R on the Cartesian coordinate plane R x R, defined by (a,b)R(c,d) iff a+d=b+c, is an equivalence relation by checking its three properties. The participants also discussed describing the equivalence class E(7,3) of this relation, which is the set of all points (c,d) that satisfy 4 = c - d and can be represented by the line y = x - 4.
  • #1
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 :cry:

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)?
 
Physics news on Phys.org
  • #2
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. :-p
 
  • #3
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)
 
Last edited by a moderator:
  • #4
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:

[tex]
f(\spadesuit) = \spadesuit^2
[/tex]
 
Last edited:
  • #5
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.
 
  • #6
Matt Grimme:

the question said "Verify that R is an equivalence relation" so R would be "="
 
  • #7
Why would "R" be "="?

"=" is an equivalence relation. Not all equivalence relations are "=".
 
  • #8
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?
 
  • #9
Looks good!
 
  • #10
oh boo-yeah, i feel good now!
Thank you!

but i still don't get what "discrie the equivelence class E(7,3)" means?
 
Last edited by a moderator:
  • #11
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!
 
  • #12
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
 
  • #13
What would happen if you wrote (7,3)R(a,b)?
 
  • #14
playboy said:
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.
 
  • #15
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?
 
  • #16
testing testing
 
  • #17
playboy said:
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?
 
  • #18
playboy said:
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!
 
  • #19
now I am happy, thanks everyone :)
 

FAQ: Proving Equivalence Relations on the Cartesian Coordinate Plane

1. What is real analysis?

Real analysis is a branch of mathematics that deals with the study of real numbers and their properties. It includes topics such as sequences, series, continuity, differentiation, integration, and limits.

2. What are relations in real analysis?

Relations in real analysis refer to the connection or association between two or more elements of a set. They can be represented as ordered pairs or sets of ordered pairs, and are used to describe the interactions between real numbers.

3. What is the difference between a function and a relation in real analysis?

A function is a type of relation where each input has only one corresponding output. In other words, for every x-value, there is only one y-value. A relation, on the other hand, may have multiple outputs for a single input. Functions are a subset of relations, and they are used to model real-world phenomena in real analysis.

4. How are relations represented in real analysis?

Relations can be represented in various ways in real analysis, such as graphs, tables, and equations. Graphs are often used to visually represent the relationship between two variables, while tables show the corresponding inputs and outputs. Equations, specifically functions, can also be used to describe relations in a mathematical form.

5. What are some applications of relations in real analysis?

Relations have many practical applications in real analysis, such as in physics, economics, and engineering. They are used to model real-world situations and make predictions or solve problems. For example, in physics, relations are used to describe the motion of objects, while in economics, they are used to analyze supply and demand.

Back
Top