Explanation
Solution
To determine if a relation is an equivalence relation, it must be Reflexive, Symmetric, and Transitive.
Analysis of R1:
The condition is (A∩BC)∪(B∩AC)=ϕ. This expression represents the Symmetric Difference (AΔB).
If AΔB=ϕ, it means A=B.
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Reflexivity: AΔA=ϕ, so AR1A is true.
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Symmetry: if A=B, then B=A, so AR1B⟹BR1A.
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Transitivity: If A=B and B=C, then A=C.
Thus, R1 is an equivalence relation (it is the identity relation).
Analysis of R2:
The condition is A∪BC=B∪AC.
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Reflexivity: A∪AC=S and A∪AC=S. Since S=S, it is reflexive.
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Symmetry: A∪BC=B∪AC is the same as B∪AC=A∪BC. It is symmetric.
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Simplifying the condition:
A∪BC=B∪AC
Taking the complement on both sides:
(A∪BC)C=(B∪AC)C
AC∩B=BC∩A
This is equivalent to B−A=A−B.
For the difference of two sets to be equal, both must be empty (i.e., A=B).
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Transitivity: Since the condition simplifies to A=B, it follows the same logic as R1.
Thus, R2 is also an equivalence relation.
Final Answer:
Both R1 and R2 are equivalence relations.
The correct option is (3).
