Let P(S) denote the power set of `S = {1,2,3,…..,10}`. Define the relation `R_(1)` and `R_(2)` on `P(S)` as `AR_(1)B` if `(A nn B^c ) uu (B nn A^c ) = phi` and `AR_2B` if `A uu B^c = B uu A^(c) , forall A,B in P(S)`. Then

Options

A. both `R_1` and `R_2` are not equivalence relations
B. only `R_2 `is an equivalence relation
C. only `R_1` is an equivalence relation
D. both `R_1` and `R_2` are equivalence relations

Answer

To solve the problem, we will analyze the two relations \( R_1 \) and \( R_2 \) defined on the power set \( P(S) \) of the set \( S = \{1, 2, 3, \ldots, 10\} \).

### Step 1: Understanding the Relation \( R_1 \)

The relation \( R_1 \) is defined as:
\[ A R_1 B \text{ if } (A \cap B^c) \cup (B \cap A^c) = \emptyset \]

This means that the union of the elements in \( A \) that are not in \( B \) and the elements in \( B \) that are not in \( A \) is empty. This can only happen if \( A \) and \( B \) are the same set. Thus, we can conclude:
\[ A R_1 B \iff A = B \]