Equivalence relation! Make formula saying that R is not an equivalence relation

Question

I don't know how to do this particular thing.

Using quantifiers and logical links as

$$and, or, =>, <=>$$

and expressions like $$x\in A, x\notin A, R(x, y), \lnot R(x, y)$$

Make formula saying that R is not an equivalence relation on set A.

Answer 1

Hint:

Recall the definition of an equivalence relation $R$ on a set $A$. $R$ will be a subset of $A \times A$. It also satisfies three properties:

• Reflexivity: Each element is related to itself, i.e. $(x,x) \in R$.
• Symmetry: If two elements are related if and only if they're related going "both ways." (Not the best words for it but the formula clarifies it.) Explicitly, $(x,y) \in R \Leftrightarrow (y,x) \in R$.
• Transitivity: Self-explanatory. $(x,y) \in R$ and $(y,z) \in R \Rightarrow (x,z) \in R$.

Begin by defining a subset $R \subseteq A \times A$ and translate the necessary properties into propositional logic and it should, within reason, all come together pretty smoothly, at least insofar as being an equivalence relation.

To be not an equivalence relation, one or more of the above properties must be false.