Sets and Relations

Let $\mathcal{C}$ be the set of all the circles in a plane. If$\mathcal{R} = \{(C_1, C_2) \in \mathcal{C} \times \mathcal{C} \mid C_1 \text{ and } C_2 \text{ intersect}\}$,then which of the following statements is TRUE?

Let $M$ be the set of all $3 \times 3$ matrices with real entries. Consider the relation $R$ on $M$ given by $R=\{(A, B) \in M \times M: \operatorname{det}(A-B)$ is an integer $\}$. Which one of the following statements is Correct?

Let $X$ be the set of all $2 \times 2$ matrices with real entries and $R \subset X \times X$ be the relation $R=$ $\{(A, B): A B=B A\}$. Which of the following statements is true?

\(\text { Define binary operation * on } Q \text { by } a * b=a+b-3 \text {. The inverse of } 2 \text { with respect to * is }\)