So I am trying to figure out which of these are equivalence relations. I think I understand the basic idea in the sense that I need to check and see if each one is reflexive, transitive, and symmetric, but I am unsure of how to proceed with these particular relations. Any ideas here?
1.) Let $A$ be a set and $B$ be a subset of $A$. Define $R$ on $\mathcal{P}(A)$ by $X ~R~ Y$ iff $X \cap B = Y \cap B$.
2.) Define $R$ on $\mathbb{N}$ by $m ~R~ n$ iff $2 | mn$.
Definitions are not ornaments. They are the tools to be used. "Reflexive, symmetric, transitive" are merely the $names$ of the tools.
$R$ is reflexive iff $(\bullet)\;\forall x\, (xRx).$
$R$ is symmetric iff $(\bullet\bullet)\;\forall x,y\,(xRy\implies yRx).$
$R$ is transitive iff $(\bullet\bullet\bullet)\;\forall x,y,z\,(\,[xRy\land yRz]\implies xRz).$
So which, if any, of $(\bullet),(\bullet\bullet),(\bullet\bullet\bullet)$ are true?
Someone else on this site said "Definitions are your friends".
For 1). Reflexive: If $x\subseteq A,$ let $y=x.$ Then $xRx \iff xRy \iff x\cap B=y\cap B\iff x\cap B=x\cap B.$
Symmetric: If $x,y\subseteq A$ let $x'=y$ and $y'=x.$ Then $$xRy\implies x\cap B=y\cap B\implies$$ $$\implies x'\cap B=y\cap B=x\cap B=y'\cap B\implies x'Ry'\implies yRx.$$
Transitive: $[xRy\land yRz]\implies$ $ [x\cap B=y\cap B\,\land \, y\cap B=z\cap B]\implies$ $\implies [x\cap B=y\cap B=z\cap B]\implies$ $ [x\cap B=z\cap B]\implies xRz.$