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

To determine whether the relations \(R_1\) and \(R_2\) are equivalence relations, we need to check if they satisfy the properties of reflexiveness, symmetry, and transitivity.

1. Checking \(R_1\):

The relation \(A R_1 B\) is defined if \((A \cap B^c) \cup (B \cap A^c) = \emptyset\). This means that \(A\) and \(B\) have the same elements (i.e., \(A = B\)). Thus, \(A R_1 B\) is essentially \(A = B\).

• Reflexivity: For any subset \(A\), \((A \cap A^c) \cup (A \cap A^c) = \emptyset\), which holds since \(B^c\) for \(B = A\) is \(\emptyset\). So, \(A R_1 A\) is true.
• Symmetry: If \(A R_1 B\), then \(A = B\), which implies \(B = A\). Therefore, \(B R_1 A\) is true.
• Transitivity: If \(A R_1 B\) and \(B R_1 C\), then \(A = B\) and \(B = C\). This implies \(A = C\), so \(A R_1 C\) is true.

Since \(R_1\) satisfies reflexivity, symmetry, and transitivity, it is an equivalence relation.

2. Checking \(R_2\):

The relation \(A R_2 B\) is defined by \(A \cup B^c = B \cup A^c\). This expression is equivalent to saying that the symmetric difference between \(A\) and \(B\) is \(\emptyset\), which implies \(A = B\).

• Reflexivity: For any subset \(A\), \(A \cup A^c = A \cup A^c\), which is always true. Hence, \(A R_2 A\) holds.
• Symmetry: If \(A R_2 B\), then \(A \cup B^c = B \cup A^c\). Interchanging \(A\) and \(B\), \(B \cup A^c = A \cup B^c\), so \(B R_2 A\) holds.
• Transitivity: If \(A R_2 B\) and \(B R_2 C\), this implies \(A = B\) and \(B = C\). Therefore, \(A = C\), so \(A R_2 C\) holds.

Since \(R_2\) also satisfies reflexivity, symmetry, and transitivity, it is an equivalence relation.

Conclusion: Both \(R_1\) and \(R_2\) are equivalence relations. The correct answer is: both \(R_1\) and \(R_2\) are equivalence relations.