Write the following English sentence as a logical expression

Question

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

Answer 1

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.

Answer 2

No, that's not quite right. I know what you try to say: for any $x$, there is an option as to whether or not $(x,x) \in R$, but while that would be understood if you say it like that in English, in logic if we say $P \lor Q$ it is not implied that we definitely sometimes have $P$ and definitely sometimes $Q$... For all we know $P$ is a tautology and $Q$ a contradiction, in which case we never have $Q$, but we do always have $P \lor Q$. Indeed, if every symmetric relations would be reflexive, then your statement would be true as well. In fact, since the consequent in your statement is a logical tautology, it will always be a true statement, no matter what!

What you want to say is that not all symmetric relations are reflexive. Or, what is the same thing, that there are symmetric relations that are not reflexive.

Also, you need to separate the bit about symmetry from the bit about reflexivity in terms of the quantified elements of A. That is, you'll want to say that there exists a relation such that it is symmetric (and now you use quantifiers for the elements of A), and it is not reflexive (and now you use a quantifier again)

Finally, there is a technical issue in that if the domain is all the relations over set A ... then how would you talk about the elements of set A? Are you sure the domain is just relations over set A?