Relation

Question

Which of the following is/are true ?

• A. $\text{R}$ is a reflexive relation on a set $\text{A}$, then $\text{R}^{n}$ is reflexive for all $n\geq0$
• B. Relation $\text{R}$ on set $A$ is reflexive if and only if inverse relation $R^{-1}$ is reflexive.
• C  Relation $\text{R}$ on set $A$ is antisymmetric if and only $R \cap R^{-1}$   is a subest of diagonal relation $\Delta = \left \{ (a,a) \; | a \in A \right \}$
• D. $M_{S\circ R} = M_R \; \odot M_S$ where $\odot$ is boolean product.

Comments

All seems true.

C. Let M be a adjacency matrix for R & M' be for $R^{-1}$ . M & M' are mutually transpose of each other, but diagonal remains same for both. Therefore $R\cap{R^{-1}}$ is nothing but $2^{n}$ numbers of diagonal fills of adjacency matrix which is sign of anti symmetric relations.

Let me know where I'm wrong if I'm.