Q46 / 79 · NEST (NISER) 2026 · Mathematics

Let $S = \{(x, y) \mid x, y \in \mathbb{N},\ 1 \leq x \leq 15,\ 1 \leq y \leq 20\}$ be a set. Let $R$ be the equivalence relation on $S$ defined by
$(x, y)\ R\ (x', y')$ if and only if $x + y = x' + y'$.
Then the number of equivalence classes of $R$ on $S$ is

  • Correct answer
    A
    $34$
  • B
    $20$
  • C
    $15$
  • D
    $35$

Solution

Given $S = \{(x, y) \mid x, y \in \mathbb{N},\ 1 \leq x \leq 15,\ 1 \leq y \leq 20\}$.

The equivalence relation $R$ is defined as $(x, y)\ R\ (x', y')$ if and only if $x + y = x' + y'$.

Each equivalence class is determined by a unique value of the sum $x + y$.

The minimum value of $x + y$ is $1 + 1 = 2$.

The maximum value of $x + y$ is $15 + 20 = 35$.

Since $x$ and $y$ take all integer values in their respective intervals, $x + y$ takes all integer values from $2$ to $35$.

The number of distinct values of $x + y$ is $35 - 2 + 1 = 34$.

Thus, the number of equivalence classes of $R$ on $S$ is $34$.

Answer: $34$

Hence, the correct answer is option: 'A'.

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