Dependence of Axioms of Equivalence Relation?

Part 1. Here's an example of a relation that is symmetric and transitive, but not reflexive.

The set is $X=\{1,2,3\}$. The relation is $$R = \Bigl\{ (2,2),\ (2,3),\ (3,2),\ (3,3)\Bigr\}.$$ Verify that $R$ is symmetric and transitive. Verify that $R$ is not reflexive. Then try to see why the alleged proof fails in this example. Use that to explain where the fallacy in the proof lies. It does not lie in taking $c$ equal to $a$.

Part 2. Think about what the fallacy is in the proof you are given; what extra hypothesis on $\sim$ would make the argument correct? The argument is fallacious because it assumes that something happens, when you have no warrant for asserting it will happen. So try to come up with some hypothesis that will guarantee this happens.

For the first part: The "proof" assumes that for $a$ there is a $b$ such that $a\sim b$. This of course is not necessarily given. The empty relation i.e. for no $a,b$ we have $a\sim b$ is transitive, symmetric but not reflexive. (This example seems to many students not very explanatory although it is the simplest example of this situation. See the example of Arturo Magidin if that helps.)

For the second part: I think the book might ask for something like this.

Prop 1': For any $a$, for which there is any $b$ such that $a\sim b$, we also have $a\sim a$.

This might seem to be an weird property, but we could also take "suffices Prop 2 and Prop 3", which wouldn't make much more sense.

Or as Arturo Magidin suggested

Prop 1'': For any $a$, there is a $b$ such that $a\sim b$

together with Prop 2 and Prop 3 implies Prop 1.

There is no difficulty in supplying a "Prop. $0$" strictly weaker than Prop. $1$ such that Prop. $0$ and Props. $2$ and $3$ imply the current Prop. $1$. Just say that for any $x$ there is a $y$ such that $x \sim y$. (Of course that $y$ might be $x$ itself.) Then Prop. $0$, together with $2$ and $3$, imply Prop. $1$. Thus Prop. $0$, $2$, and $3$ give an alternate axiomatization. Perhaps this is the intended answer. Or not.