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Equivalence Relation

If A and B ar...

Question

If A and B are equivalence relations defined on a set C,then explain how A intersection B is an equivalence relation.kindly do not post any link.

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Solution

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$It is given that A and B are relations on set C . ∴ A \subseteq C \times C and B \subseteq C \times C \Rightarrow A \cap B \subseteq C \times C \Rightarrow A \cap B is also a relation on C . Now, we shall show that it is an equivalence relation on C . We observe that the following properties of relation A \cap B : Reflexivity : Let a be an arbitrary element of C . then, a \in C \Rightarrow (a, a) \in A and (a, a) \in B [∵ A and B are reflexive] \Rightarrow (a, a) \in A \cap B Thus, (a, a) \in A \cap B for all a \in C . So, A \cap B is a reflexive relation on C . Symmetry : Let a, b \in C such that (a, b) \in A \cap B . Then, (a, b) \in A \cap B \Rightarrow (a, b) \in A and (a, b) \in B \Rightarrow (b, a) \in A and (b, a) \in B [∵ A and B are symmetric] \Rightarrow (b, a) \in A \cap B thus, (a, b) \in A \cap B \Rightarrow (b, a) \in A \cap B for all (a, b) \in A \cap B So, A \cap B is symmetric on C . Transitivity : Let a, b, c \in C such that (a, b) \in A \cap B and (b, c) \in A \cap B . Then, (a, b) \in A \cap B and (b, c) \in A \cap B \Rightarrow \{(a, b) \in A and (a, b) \in B\} and \{(b, c) \in A and (b, c) \in B\} \Rightarrow \{(a, b) \in A, (b, c) \in A\} and \{(a, b) \in B, (b, c) \in B\} \Rightarrow (a, c) \in A and (a, c) \in B [\begin{matrix} ∵ A and B are transitive . \\ ∴ (a, b) \in A and (b, a) \in A \Rightarrow (a, c) \in A \\ (a, b) \in B and (b, a) \in B \Rightarrow (a, c) \in B \end{matrix}] \Rightarrow (a, c) \in A \cap B Thus, (a, b) \in A \cap B and (b, c) \in A \cap B \Rightarrow (a, c) \in A \cap B So, A \cap B is transitive on A . hence, A \cap B is an equivalence relation on C .$
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