1. Given set and relations

Let $S=\{1,2,3,\dots,10\}, \qquad P(S)=\text{power set of }S.$

We define relations on $P(S)$:

• $A\,R_1\,B$ if $\left(A\cap B^c\right)\cup\left(B\cap A^c\right)=\emptyset$
• $A\,R_2\,B$ if $A\cup B^c=B\cup A^c$

We must check whether $R_1$ and $R_2$ are equivalence relations.

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2. Analyze $R_1$

We know $\left(A\cap B^c\right)\cup\left(B\cap A^c\right)$ is the symmetric difference of $A$ and $B$, i.e. $A\triangle B=(A\cap B^c)\cup(B\cap A^c).$

So the condition for $R_1$ is $A\triangle B=\emptyset.$

But $A\triangle B=\emptyset \iff A=B.$

Hence, $A\,R_1\,B \iff A=B.$

So $R_1$ is just the equality relation on $P(S)$.

Now check equivalence properties:

(i) Reflexive

For every $A\in P(S)$, $A=A,$ so $A\,R_1\,A$.

(ii) Symmetric

If $A\,R_1\,B$, then $A=B$. Hence $B=A$, so $B\,R_1\,A$.

(iii) Transitive

If $A\,R_1\,B$ and $B\,R_1\,C$, then $A=B$ and $B=C$. Therefore $A=C$, so $A\,R_1\,C$.

Thus, $R_1$ is an equivalence relation.

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3. Analyze $R_2$

Given $A\cup B^c=B\cup A^c.$ We simplify this relation.

A standard way is to compare membership elementwise.

Take any $x\in S$. Let

• $a=1$ if $x\in A$, else $0$
• $b=1$ if $x\in B$, else $0$

Then:

• $x\in A\cup B^c$ means $a\lor \neg b$
• $x\in B\cup A^c$ means $b\lor \neg a$

We need $a\lor \neg b = b\lor \neg a$ for every element $x$.

Check all four cases:

Equality holds exactly when $a=b$.

Therefore, for every $x\in S$, $x\in A \iff x\in B.$ Hence, $A=B.$

So, $A\,R_2\,B \iff A=B.$

Thus $R_2$ is also the equality relation on $P(S)$.

Now check equivalence properties:

(i) Reflexive

For every $A$, $A\cup A^c=S=A\cup A^c,$ so $A\,R_2\,A$.

(ii) Symmetric

If $A\,R_2\,B$, then $A\cup B^c=B\cup A^c.$ Reversing sides gives $B\cup A^c=A\cup B^c,$ so $B\,R_2\,A$.

(iii) Transitive

Since $A\,R_2\,B \iff A=B$, if $A\,R_2\,B$ and $B\,R_2\,C$, then $A=B$ and $B=C$, hence $A=C$, so $A\,R_2\,C$.

Thus, $R_2$ is an equivalence relation.

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4. Evaluate options

• A: only $R_2$ is an equivalence relation ❌
• B: both $R_1$ and $R_2$ are not equivalence relations ❌
• C: both $R_1$ and $R_2$ are equivalence relations ✅
• D: only $R_1$ is an equivalence relation ❌

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5. Final answer

Both $R_1$ and $R_2$ are equivalence relations.