# Reflexive, Symmetric, Transitive - Prove related problem

## Homework Statement

Let A=RxR=the set of all ordered pairs (x,y), where x and y are real numbers. Define relation P on A as follows: For all (x,y) and (z,w) in A, (x,y)P(z,w) iff x-y=z-w

## Homework Equations

R is reflexive if, and only if, for all x ∈ A,x R x.
R is symmetric if, and only if, for all x,y∈A,if xRy then yRx.
R is transitive if, and only if, for all x,y,z∈A, if xRy and yRz then xRz.

## The Attempt at a Solution

I am supposed to prove that P is reflexive, symmetric and transitive.
To show if P is reflexive do I just state that since y-x=w-z then L is reflexive
To show it is symmetric do I just state y-x=w-z and x-y=z-w, since they are equivalent then they are symmetric
To show if Transitive if (the book gives this answer for a similar problem) since x-y and z-w are integers then x-z=(x-y) + (x-z) is the sum of two integers. Therefore x-z is an integer. Assuming this is correct, are my two answers for reflexive and symmetric incorrect?

Furthermore, how would I list five elements in [(2,6)] or [(5,5)]?

Thanks for the help! :)

Mark44
Mentor

## Homework Statement

Let A=RxR=the set of all ordered pairs (x,y), where x and y are real numbers. Define relation P on A as follows: For all (x,y) and (z,w) in A, (x,y)P(z,w) iff x-y=z-w

## Homework Equations

R is reflexive if, and only if, for all x ∈ A,x R x.
R is symmetric if, and only if, for all x,y∈A,if xRy then yRx.
R is transitive if, and only if, for all x,y,z∈A, if xRy and yRz then xRz.

## The Attempt at a Solution

I am supposed to prove that P is reflexive, symmetric and transitive.
To show if P is reflexive do I just state that since y-x=w-z then L is reflexive
What do you get for (x, y)P(x, y)?
eseefreak said:
To show it is symmetric do I just state y-x=w-z and x-y=z-w, since they are equivalent then they are symmetric
You need to show that (x, y) P (z, w) is the same as (z, w) P (x, y)
eseefreak said:
To show if Transitive if (the book gives this answer for a similar problem) since x-y and z-w are integers then x-z=(x-y) + (x-z) is the sum of two integers.
You can't assume the numbers are integers. The relations are defined on sets of real ordered pairs.
eseefreak said:
Therefore x-z is an integer. Assuming this is correct, are my two answers for reflexive and symmetric incorrect?

Furthermore, how would I list five elements in [(2,6)] or [(5,5)]?
Use the definition of your relation.
eseefreak said:
Thanks for the help! :)

Last edited:
What do you get for (x, y)P(x, y)?
You need to show that (x, y) P (z, w) is the same as (z, w) P (x, y)

Use the definition of your relation.

How can I show that they are the same? It seems easy to show they are the same if there are variables in place but how would I show they are the same for all real numbers?

btw thanks Mark44 for the help

What is (z,w)P(x,y)? my guess would be having to prove that x-y=y-x

Fredrik
Staff Emeritus