Question

Let \(P(S)\) denote the power set of \(S = \{1, 2, 3, \ldots, 10\}\). Define the relations \(R_1\) and \(R_2\) on \(P(S)\) as \(A R_1 B\) if \[(A \cap B^c) \cup (B \cap A^c) = ,\]and \(A R_2 B\) if\[A \cup B^c = B \cup A^c,\]for all \(A, B \in P(S)\). Then:

• A. both\(R_1\) and \(R_2\) are  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 not equivalence relations

Answer 1

The Correct Option is A

Let \(S = \{1, 2, 3, \ldots, 10\}\), and \(P(S)\) denote the power set of \(S\).
Relation \(R_1: A R_1 B\) if
\[(A \cap B^c) \cup (B \cap A^c) = \varnothing.\]
This implies \(A = B\).
Step 1: Check Reflexivity
For any \(A \in P(S)\),
\[(A \cap A^c) \cup (A^c \cap A) = \varnothing.\]
Thus, \(AR_1A\). Therefore, \(R_1\) is reflexive.
Step 2: Check Symmetry
If \(AR_1B\), then \((A \cap B^c) \cup (B \cap A^c) = \varnothing\). This implies \(A = B\), and thus \(BR_1A\). Therefore, \(R_1\) is symmetric.
Step 3: Check Transitivity
If \(AR_1B\) and \(BR_1C\), then \(A = B\) and \(B = C\), which implies \(A = C\). Therefore, \(R_1\) is transitive.
Thus, \(R_1\) is an equivalence relation.
Relation \(R_2\): \(A R_2 B\) if
\[A \cup B^c = B \cup A^c.\]
Step 1: Check Reflexivity
For any \(A \in P(S)\),
\[A \cup A^c = A \cup A^c.\]
Thus, \(AR_2A\). Therefore, \(R_2\) is reflexive.
Step 2: Check Symmetry
If \(AR_2B\), then \(A \cup B^c = B \cup A^c\). By symmetry of the union operation, \(B \cup A^c = A \cup B^c\), which implies \(BR_2A\). Therefore, \(R_2\) is symmetric.
Step 3: Check Transitivity
If \(AR_2B\) and \(BR_2C\), then:
\[A \cup B^c = B \cup A^c,\]
\[B \cup C^c = C \cup B^c.\]
Using substitution, we get:
\[A \cup C^c = C \cup A^c,\]
which implies \(AR_2C\). Therefore, \(R_2\) is transitive.
Thus, \(R_2\) is an equivalence relation.
Conclusion: Both \(R_1\) and \(R_2\) are equivalence relations