Let a set $A = A_1 \cup A_2 \cup ..... \cup A_k$, where $A_i \cap A_j = \phi$ for $i \neq j$, $1 \le j, j \le k$. Define the relation R from A to A by $R = \{(x, y) : y \in A_i$ if and only if $x \in A_i, 1 \le i \le k\}$. Then, R is :
(A)
reflexive, symmetric but not transitive.
(B)
reflexive, transitive but not symmetric.
(C)
reflexive but not symmetric and transitive.
(D)
an equivalence relation.
MEDIUM
Correct Answer:D
Explanation
$R = \{(x, y) : y \in A_i, iff x \in A_i, 1 \le i \le k\}$
(1) Reflexive: $(a, a) \Rightarrow a \in A_i$ iff $a \in A_i$
(2) Symmetric: $(a, b) \Rightarrow a \in A_i$ iff $b \in A_i$. $(b, a) \in R$ as $b \in A_i$ iff $a \in A_i$
(3) Transitive: $(a, b) \in R$ & $(b, c) \in R$. $\Rightarrow a \in A_i$ iff $b \in A_i$ & $b \in A_i$ iff $c \in A_i$. $\Rightarrow a \in A_i$ iff $c \in A_i$. $\Rightarrow (a, c) \in R$.
Therefore, the relation is an equivalence relation.