Question:

Given: Set = \{1, 2, 3, 4, 5\}$ and relation = \{(a, b) : |a - b| ext{ is even}\}$.
Step 1: Test for Reflexivity:
For any element \in A$, $|a - a| = 0$, which is an even integer.
Therefore, \in R$ for all \in A$.
Thus, $ is reflexive. [0.5 Mark]
Step 2: Test for Symmetry:
Let \in R$. Then $|a - b|$ is even.
Since $|b - a| = |-(a - b)| = |a - b|$, it follows that $|b - a|$ is also even.
Therefore, \in R$.
Thus, $ is symmetric. [0.5 Mark]
Step 3: Test for Transitivity:
Let \in R$ and \in R$.
Then $|a - b|$ is even and $|b - c|$ is even.
$\implies (a - b)$ is even and $ is even.
Sum of two even integers is even:
35529(a - b) + (b - c) = a - c ext{ is even} \implies |a - c| ext{ is even}35529
Therefore, \in R$.
Thus, $ is transitive. [0.5 Mark]
Conclusion: Since $ is reflexive, symmetric, and transitive, $ is an equivalence relation. [0.5 Mark]

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🎯 Official CBSE Marking Scheme:
Proving reflexivity: 0.5 Mark
Proving symmetry: 0.5 Mark
Proving transitivity: 0.5 Mark
Concluding equivalence relation: 0.5 Mark