Let $A = \{1,2,3,4\}$ and let $f: \mathcal{P}(A)\to\mathbb{N}$ be the function defined by saying that $f(X)$ is the sum of elements of $X$, for each $X\in \mathcal{P}(A)$. (If $X = \emptyset$, then by convention we say that $f(X)=0$). Define the relation $\sim$ on $\mathcal{P}(A)$ as follows:
$X\sim Y$ if and only if $f(X) > f(Y)$ or $X = Y$.
Write down whether $\sim$ is reflexive, symmetric, antisymmetric, transitive.
The part where I am struggling is to prove reflexivity.
Reflexive: $a\sim a$ for all $a$. So that means $f(X) > f(Y)$ is false, since $f(X) > f(X)$ is false.
But $X = Y$ part is true since, $X = X$ for all $X \in\mathcal P(A)$.