I understand how to prove basic problems ($(X, Y ) \in R_1$ iff $X^2=Y^2$, for example) on equivalence relations but I have some difficulty solving equivalence relations involving sets and power sets.
Is P(S) the domain and codomain here for X, Y?
Any help would be appreciated.
Let $S$ be a set with at least two elements. Consider the following three relations on $\mathcal P(S)$. Determine which of the properties - reflexivity, symmetry, antisymmetry, transitivity - each of relations possesses. Give a proof or counterexample as appropriate.
(a) $(X, Y ) \in R_1$ if and only if $X \subseteq Y$.
(b) $(X, Y ) \in R_2$ if and only if $X$ is strictly a subset of $Y$ ($X$ does not equal to $Y$).
(c) $(X, Y ) \in R_3$ if and only if $X \cap Y = \emptyset$.
Solution for (a):
Let $X, Y, Z \in \mathcal P(S)$.
Reflexivity is satisfied, since obviously $X \subseteq X$
Symmetry is not satisfied. We know $S$ has at least two elements, so we can choose two distinct elements $a, b \in S$. Then $\{a\} \subseteq \{a,b\}$ but $\{a,b\} \not \subseteq \{a\}$.
Transitivity is satisfied: Let $X \subseteq Y \subseteq Z$. Now, if $X=\emptyset$, then $X \subseteq Z$. So let $X \neq \emptyset$ and let $x \in X$. Then $x$ is in $Y$ but since $Y \subseteq Z$, we also have $x \in Z$. That shows $X\subseteq Z$.
Can you do the rest now?