Is the subset relation on the powerset of a set, with qualification, reflexive?

Question

I was wondering if the subset relation is reflexive?

$R = \{(X, Y ) \in P(A)^2\mid X\subseteq Y \text{ and } X \neq Y \}$

I assumed they it was reflexive since for all $X \in P(A), X \subseteq X$ is true. So does that mean the relation is reflexive? Or do I need to add additional proof?

Answer 1

The relation cannot be reflexive. $(X, X)\in R$ if and only if both the following statements are true:

$$X\subseteq X\tag{1}$$ $$X \neq X\tag{2}$$

$(1)$ is true for all $X \in \mathcal P(A)$. $(2)$ is false for all $X \in \mathcal P(A)$.

Hence, for all $X \in \mathcal P(A)$, $(X, X) \notin R$.