Let $\Bbb R$ be the set of real numbers. For $x, y \in \Bbb R$, $x \sim y$ iff $x + y \in \Bbb Z$, i.e., if the sum $x + y$ is an integer. Determine:
(i) Whether or not the relation $\sim$ is reflexive;
(ii) Whether or not the relation $\sim$ is symmetric;
(iii) Whether or not the relation $\sim$ is anti-symmetric;
(iv) Whether or not the relation $\sim$ is transitive;
(v) Whether or not the relation $\sim$ is an equivalence relation;
(vi) Whether or not the relation $\sim$ is a partial order.
Justify answers.
I understand the concepts but I am unsure on how to answer with wording and justify the answers correctly.
1) No the relation is not reflexive since $x+x$ may not be an integer (for a counter-example, just take $x=0.2$)
2) Yes it is symmetric since if $x\sim y$, it means $x+y$ is an integer. So, $y+x$ is also an integer. So, $y\sim x$
3) No it is not anti-symmetric since $0.5+1.5$ is integer but $0.5\neq 1.5$
4) No it is not transitive since $0.1+1.9$ is integer and $1.1+1.9$ is integer but $1.1+0.1$ is not an integer.
5) No since relation must be R-S-T for this
6) No since relation must be R-AS-T for this