Indicate if the following relation is is reflexive, symm...

[stiffy]

Q: Is the following relation reflexive, symmetric, or transitive?

R, where (x,y)R(z,w) iff x+z ? y+w, on the set RxR.

I am not sure how to treat the relation of two ordered pairs.
Here is what I did.

Reflexive: Suppose (x,y)?R. Then x+y?x+y since x+y=x+y. This, (x,y)R(x,y). So, R is reflexive.

Symmetric: Suppose (x,y)R(z,w) and (z,w)R(x,y). Then, x+z ? y+w ? y+w ? x+z so, x+z ? y+w ? x+z which implies x+z=y+w. Therefore, R is symmetric. (I am pretty sure this is wrong, because I don't think I can just stick the relation "?" in the middle of the two relations. If not, I cannot state R is symmetric.)

Transitive: Suppose (x,y)R(z,w) and (z,w)R(u,v) where (u,v) are arbitrary coordinates in R. Then, x+z ? y+w and z+u ? w+v. so, we can rewrite this as x+z+u ? y+w+v. So, R is transitive.

Thats me attempt at it.

Thank you
--Dan

 

[pka]

Q: Is the following relation reflexive, symmetric, or transitive?
R, where (x,y)R(z,w) iff x+z ? y+w, on the set RxR.
Symmetric: Suppose (x,y)R(z,w) and (z,w)R(x,y). Then, x+z ? y+w ? y+w ? x+z so, x+z ? y+w ? x+z which implies x+z=y+w. Therefore, R is symmetric.

\(\displaystyle (a,b)\mathcal{R}(c,d)\; \Rightarrow \;a + c \leqslant b + d \Leftrightarrow c + a \leqslant d + b\; \Rightarrow \;(c,d)\mathcal{R}(a,b)\).
That proves it is symmetric.

 

[stiffy]

Oh, so to show its reflexive I show both first coordinates are related, as in (x,z)R(x,z)?

 

[pka]

Oh, so to show its reflexive I show both first coordinates are related, as in (x,z)R(x,z)?

Is it true that \(\displaystyle (3,1)\mathcal{R} (3,1)\)?
If not it is not reflexive.

 

[stiffy]

so,....
We can see R is not reflexive by counter example. (3,1)R(3,1) implies 6?2. This does not hold obviously.
R is symmetric.
R is not transitive. We can find any two relations defined in my original post that hold, but we are not guaranteed there exists an ordered pair shared by these two relations such that the ordered pair will relate the two relations under our definition.