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
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! :)