Write the following English sentence as a logical expression I need some help double-checking a solution to a logic problem. The questions is:
Let the universe of discourse be the set of relations on the set A. Write the following English sentence as a logical expression, using quantifiers, logical expressions, and set notation.
a) A symmetric relation on a set A is not necessarily reflexive.
My solution is $$\forall x \in A,\forall y \in A\Big(\big((x,y)\in R \to (y,x) \in R\big) \to \big((x,x)\in R)\lor ((x,x)\not\in R)\big)\Big)$$
Thank you

What you need, and your second attempt is a great improvement over the first, is to check your first connective, associated with the existence of a relation:
$$\exists R\Big(\big((R\subseteq A\times A)\land \forall x, y\in A((x,y)\in R \to (y, x) \in R)\big) \land \exists z \in A((z, z)\notin R)\Big)$$
I used more parentheses than you did, to help define the scope of each quantified variable.  I also changed the connective between $R\subseteq A\times A$ and $\forall x, y \in A(.....)$, Because R being simply a subset of $A\times A$ does not imply it is symmetric.  With Using $\land$ there, we are describing R as a subset of $A\times A$, AND, also, as a  symmetric relation.
