Sets, Relations & Functions
Equivalence Relation / Statements
nta_pyq_2025_apr
Grade 11
Question:
Let $X = \mathbb{R} \times \mathbb{R}$. 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 $y = x$. Choose the correct option:
Both I and II are false
I is true but II is false
Both I and II are true
I is false but II is true
Step-by-Step Solution
Key Concept: Verify each property of $R$: reflexive ($b=b$ ✓), symmetric (if $b_1=b_2$ then $b_2=b_1$ ✓), transitive (if $b_1=b_2$ and $b_2=b_3$ then $b_1=b_3$ ✓). For Statement II, $S=\{(x,y): y=b\}$ is a horizontal line, not parallel to $y=x$.
R is reflexive, symmetric, transitive $\to$ equivalence relation (Statement I: True). $S = \{(x,y): y = b\}$ is a horizontal line, not parallel to $y = x$ (Statement II: False). Answer: Statement I true, Statement II false.
Correct Answer: Statement I is true but Statement II is false