Restriction of an equivalence relation on a subset.

Question

If we have an equivalence relation defined on a set E and S its subset. Is the relation defined on S is also an equivalence one? Thank you for your answers.

Answer 1

If your new equivalence relation $\underset{S}{\sim}$ is a constriction from the whole set $E$, then the answer is yes. Equivalence relation have to satisfy 3 axioms:

1. Reflexivity. $a \underset{S}{\sim} a$. It is true for new relation because of the fact that we can look at the element $a$ like an element of the whole $E$ and use fact that $a \underset{E}{\sim} a$.

2. Symmetry. If $a \underset{S}{\sim} b$ then $b \underset{S}{\sim} a$. Using reasoning from 1 we conclude that this is true.

3. Transitivity. If $a \underset{S}{\sim} b$ and $b \underset{S}{\sim} c$ then $a \underset{S}{\sim} c$. Because of the fact that $a, b, c \in S \subset E$ then we conclude that this is also true for a restricted relation.