Let $S=\{(a, b) \mid a, b \in \mathbb{Z}\}$. Let $R$ be the equivalence relation on $S$ defined by $(a, b) R(c, d)$ if $a^2+b^2=c^2+d^2$. For $(a, b) \in S$ let $F(a, b)$ denote the equivalence class $\{(c, d) \in S \mid(a, b) R(c, d)\}$ of $(a, b)$. Then

• Correct answer
  A

  there exists $(a, b) \in S$ such that $F(a, b)$ has only one element.
• Correct answer
  B

  there exists $(a, b) \in S$ such that $F(a, b)$ has exactly 4 elements.
• C

  there exists $(a, b) \in S$ such that $F(a, b)$ has exactly 6 elements.
• D

  there exists $(a, b) \in S$ such that $F(a, b)$ has infinitely many elements.

Hence, the correct answer is options: 'A', 'B'.