For a given universe $U$ and a fixed subset $C$ of $U$, we define $R$ on $P(U)$ ("parts of $U$") as follows: for any $A,B⊆U$ we have ARB if and only if $A\cap C = B\cap C$
Considering that the definition of properties are:
- Reflective: $\forall x\in aRa$
- Symmetrical: $\forall a, b \in A / a R b \Rightarrow b R a $
- Antisymmetric: $\forall a, b \in A / [a R b \wedge b R a] \Rightarrow a = b$
- Transitive: $\forall a, b, c \in A / [a R b \wedge b R c] \Rightarrow a R c$
Here are a couple of hints to get you started.
To show that $R$ is reflexive, you must show that for any $A\subseteq U$ we have $A R A$, i.e., $A\cap C=A\cap C$; is that always true?
To show that $R$ is antisymmetric, you must show that for any $A,B\subseteq U$, if $A R B$ and $B R A$, then $A=B$, i.e., if $A\cap C=B\cap C$ and $B\cap C=A\cap C$, then $A=B$. What happens if, say, $C$ is a proper subset of $U$, $A=C$, and $B=U$? Are $A\cap C$ and $B\cap C$ equal in that case? Are $A$ and $B$ equal? Can you describe a specific example of this form?
