The below picture is a problem involving relations from a review set. I have only seen problems with relations involving matrices and solving for the properties of reflexive, symmetric, and transitive.
So determining if a relation is anti-symmetric and quite simply approaching parts B and C where you're given the relation not already in a matrix is problematic.
Is there a way to transform the given relation in B and C to a matrix? And how, from a matrix, can we determine if the relation is anti-symmetric?
Here is the matrix setup for B.
\begin{bmatrix}X&1&2&3&4&5&6\\1&?&?&?&?&?&?\\2&?&?&?&?&?&?\\3&?&?&?&?&?&?\\4&?&?&?&?&?&?\\5&?&?&?&?&?&?\\6&?&?&?&?&?&?\end{bmatrix}
See my comments on the question for examples demonstrating the difference between symmetric and antisymmetric relations.
Given a relation $R$ on a set $X$ with $|X|=n<\infty$, say, then $R$ is equivalent to an $n\times n$ matrix $\mathcal{R}$ with entries in $\{0, 1\}$ (or $\{\text{false, true}\}$ if you prefer), where, if one labels the rows & columns according to the elements of $X$, the entry
$$\mathcal{R}_{ij}:=\begin{cases} 0\,\text{(false)} & \text{if not } \quad iRj, \\ 1\,\text{(true)} & \text{if }\quad iRj. \end{cases}$$
As far as I am aware, there is no easy way to see if $R$ is (anti)symmetric on $X$, given only $\mathcal{R}$, but I could be wrong.