Question ID: #328
Let $X=R\times R$. Define a relation $R$ on $X$ as:
$$(a_{1},b_{1})R(a_{2},b_{2}) \Leftrightarrow b_{1}=b_{2}.$$
Statement-I: $R$ is an equivalence relation.
Statement-II: For some $(a, b) \in X$, the set $S=\{(x,y)\in X:(x,y)R(a,b)\}$ represents a line parallel to $y=x.$
In the light of the above statements, choose the correct answer from the options given below:
- (1) Both Statement-I and Statement-II are false.
- (2) Statement-I is true but Statement-II is false.
- (3) Both Statement-I and Statement-II are true.
- (4) Statement-I is false but Statement-II is true.
Solution:
**Check Statement-I:**
A relation is an equivalence relation if it is Reflexive, Symmetric, and Transitive.
1. **Reflexive:** Let $(a, b) \in X$.
Since $b = b$, we have $(a, b)R(a, b)$.
Thus, $R$ is reflexive.
2. **Symmetric:** Let $(a_1, b_1)R(a_2, b_2)$.
This implies $b_1 = b_2$.
Clearly, $b_2 = b_1$, which implies $(a_2, b_2)R(a_1, b_1)$.
Thus, $R$ is symmetric.
3. **Transitive:** Let $(a_1, b_1)R(a_2, b_2)$ and $(a_2, b_2)R(a_3, b_3)$.
This implies $b_1 = b_2$ and $b_2 = b_3$.
Therefore, $b_1 = b_3$, which implies $(a_1, b_1)R(a_3, b_3)$.
Thus, $R$ is transitive.
Since $R$ satisfies all three conditions, Statement-I is **True**.
**Check Statement-II:**
The set $S$ is defined as $S = \{(x, y) \in X : (x, y)R(a, b)\}$.
Using the definition of $R$, $(x, y)R(a, b) \Leftrightarrow y = b$.
So, $S = \{(x, b) : x \in R\}$.
Geometrically, $y = b$ represents a straight line parallel to the x-axis (horizontal line).
The line $y = x$ has a slope of 1.
A horizontal line ($slope = 0$) is **not** parallel to $y = x$ ($slope = 1$).
Thus, Statement-II is **False**.
Ans. (2)
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