Binary Relations - Reflexive, Symmetric, Transitive and anti symmetric

Question

$R$ is defined on $P(N) − \{\varnothing\}$ by $ARB$ if and only if $A \cap B \ne \varnothing$

Identify if the relation is reflexive, symmetric, transitive and anti symmetric

Finding it hard to work with this one.

if $P(A)$ is $\{\}$ and $\{A\}$ then the intersection would be $\{\}, \{A\}$ making it reflexive?

Answer 1

Well the relation is obviously reflexive. Because if we take the set $A$ then $A \cap A = A \neq \emptyset$ and of course we took $A$ to be not the empty set in the first place.

Now the relation is also obviously symmetric because if $A \cap B \neq \emptyset$ then $B \cap A \neq \emptyset$ is true.

Now is the relation transitive ?

if we have $A \cap B \neq \emptyset$ and $B \cap C \neq \emptyset$ Does this imply that $A \cap C \neq \emptyset$

It's not transitive and I will let you think why it's not.