Sets, Relations & Functions
Properties of Relations
nta_pyq_2025_apr
Grade 11
Question:
The relation $R = \{(x,y) : x, y \in \mathbb{Z} \text{ and } x + y \text{ is even}\}$ is:
reflexive and symmetric but not transitive
an equivalence relation
symmetric and transitive but not reflexive
reflexive and transitive but not symmetric
Step-by-Step Solution
Key Concept: Check all three properties: $x+x=2x$ (even) $\to$ reflexive; $x+y$ even $\Rightarrow y+x$ even $\to$ symmetric; $x+y$ and $y+z$ even $\Rightarrow x+z=(x+y)+(z+y)-2y$ even $\to$ transitive.
Reflexive: $x+x=2x$ (even) ✓. Symmetric: $x+y$ even $\Rightarrow y+x$ even ✓. Transitive: $x+y$ even and $y+z$ even $\Rightarrow x+2y+z$ even $\Rightarrow x+z$ even ✓. $R$ is an equivalence relation.
Correct Answer: an equivalence relation