What properties of relations does this satisfy?

Question

Define the relation $M(A, B) :\ A \cap B = \varnothing$, where the domains for $A$ and $B$ are all subsets of $\mathbb Z$. Which properties does the relation $M$ satisfy?

Answer 1

Of these 5 there is indeed only one satisfied, but not the irreflexive property: it's the symmetric one.

We have $A \cap A = \varnothing$ iff $A=\varnothing$, which means that $M$ is neither reflexive (take $A=\mathbb Z$) nor irreflexive (take $A=\varnothing$).

It is not transitive: take $A=\{1\}$, $B=\{2\}$ and $C=\{1\}$.

It is symmetric, since $B\cap A=A\cap B=\varnothing$ (assuming $A\cap B=\varnothing$).

It is not antisymmetric: take $A=\{1\}$ and $B=\{2\}$.