Let $G=\{(x,y)\in \mathbb{Z}\times\mathbb{Z}:x+y<4\}$ Is $G$ irreflexive?
I know $<$ is an irreflexive relation. But how to show it?
How do I do this?
The Reflexive and Irreflexive property always gets me. We might as well look at other properties too. I can show G is transitive If $x+y<4$ and $y+z<4$. Then $x+z<4$
Since $(1,1) \in G,$ so $G$ is not irreflexive.
Since $(3,3)\notin G,$ so $G$ is not reflexive either.
Since $(1,2),(2,1) \in G$ but $1\neq 2$ so $G$ is not antisymmetric.
$G$ is symmetric because $x+y <4 \implies y+x <4.$
Since $(3,0),(0,3)\in G$ but $(3,3) \notin G,$ so $G$ is not transitive.
You might want to revisit the definitions if you're having trouble in proving these properties.