$R$ 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?

  • $\begingroup$ I'm not sure what you mean in your last sentence. Would you mind clarifying it? $\endgroup$
    – jgon
    Apr 29, 2015 at 7:14
  • $\begingroup$ do you mean equal or not equal $\emptyset$ $\endgroup$
    – alkabary
    Apr 29, 2015 at 7:25
  • $\begingroup$ From your last sentence, I would suggest that the first thing you need to do is make sure you know exactly what the term "reflexive" means. $\endgroup$
    – David
    Apr 29, 2015 at 7:30

1 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.


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