Question

If A=∅
1. Is it an equivalence relation?
2. Is it antisymmetric?

Answer 1

Question 1

Let $A=\emptyset$.

1. Is it an equivalence relation?

2. Is it antisymmetric?

Solution

1. By definition, a relation $A$ on $X$ is an equivalence relation if and only if it is reflexive, i. e. for all $x\in X$:

$(x,x)\in A,$

symmetric, i. e. for all $x,y\in X$:

$(x,y)\in A\Rightarrow(y,x)\in A$

and transitive, i. e. for all $x,y,z\in X$:

$(x,y),(y,z)\in A\Rightarrow(x,z)\in A.$

So, if $X\neq\emptyset$, then $A=\emptyset$ cannot be an equivalence relation, because it is not reflexive. Now suppose $X=\emptyset$. Then $A=\emptyset=X\times X$ is, obviously, reflexive, symmetric and transitive, because all the premises of the implications above are always false (there are no elements in $X$) and hence the implications are true.

2. A relation $A$ on $X$ is antisymmetric if for all $x,y\in X$:

$(x,y),(y,x)\in A\Rightarrow x=y.$

Clearly, if $A=\emptyset$, then the premise of the implication is false, so the implication is true and thus $A$ is antisymmetric.

Answer:

1. If $X=\emptyset$, then $A$ is an equivalence relation on $X$, otherwise it is not an equivalence relation on $X$;

2. $A$ is antisymmetric.

∎