Is the following an equivalence relation on $\mathbb{R} \times \mathbb{R} $?

Question

Define $(x,y) R (z,w) $ iff $x + z \leq y + w $. Is $R$ an equivalence relation on $\mathbb{R} \times \mathbb{R} $?

So far I got reflexivity and symmetry which are obvious. However, I am stuck on transitivity. It seems to me that transitivity does not hold, but I am unable to find a counterexample.

score 4 · Accepted Answer · answered Mar 11 '15 at 07:28

4

The definition of reflexivity is $$tRt\quad\forall t\in \mathbb{R}^2$$ Therefore if $t=(x,y)$ $$x+x\le y+y$$ or $$x\le y$$ clearly is not true when $x\gt y$. So the relation is not reflexive.

answered Mar 11 '15 at 07:28

Fermat

5,230

is the relation transitive ? – Mar 11 '15 at 08:38
No. $(3,1)R(-2,5)$ and $(-2,5)R(3,2)$ but it does not imply that $(3,1)R(3,2)$ since $6\lt 3$ is false. – Fermat Mar 11 '15 at 08:47

Is the following an equivalence relation on $\mathbb{R} \times \mathbb{R} $?

1 Answers1