Determine if these are equivalence relations

Question

I would appreciate if someone could go through the task and the answers I've got and check if I've done it correct, if not please correct me.

Here is the task:

Below we have listed some relationships over The set of $\{a, b, c, d, e\}$. For each of these determine whether it is an equivalence relation, and in that case find $[a]$, which is equivalence class to $a$.

1. $\{\langle a,a\rangle, \langle b,b \rangle, \langle c,c\rangle, \langle d,d\rangle\}$

2. $\{\langle a,a\rangle, \langle b,b\rangle, \langle c,c\rangle, \langle d,d\rangle, \langle e,e\rangle, \langle a,b\rangle, \langle b,a\rangle\}$

3. $\{\langle a,a\rangle, \langle b,b\rangle, \langle c,c\rangle, \langle d,d\rangle, \langle e,e\rangle, \langle b,c\rangle, \langle b,d\rangle, \langle c,b\rangle, \langle d,b\rangle\}$

4. $\{\langle a,a\rangle, \langle b,b\rangle, \langle c,c\rangle, \langle d,d\rangle, \langle e,e\rangle, \langle a,c\rangle, \langle b,c\rangle\}$

5. $\{\langle a,a\rangle, \langle b,b\rangle, \langle c,c\rangle, \langle d,d\rangle, \langle e,e\rangle, \langle b,d\rangle, \langle d,b\rangle\}$

6. $\{\langle a,a\rangle, \langle b,b\rangle, \langle c,c\rangle, \langle d,d\rangle, \langle e,e\rangle, \langle b,d\rangle, \langle d,b\rangle, \langle a,c\rangle, \langle c,a\rangle\}$

Here are my answers:

1. $[a] = \{a\}$
2. $[a]=\{a, b\}$
3. $[a]=\{a\}$
4. $[a]=\{a, c\}$
5. $[a]=\{a\}$
6. $[a]=\{a, c\}$

If I'm correct $(1)$, $(3)$, $(5)$ and $(4)$ and $(6)$ are equivalence relations.

Thanks a lot for your help.

Answer 1

A relation $R$ is an equivalence relation if and only if it is reflexive AND symmetric AND transitive.

Equivalence classes are determined by an equivalence relation. It makes no sense to speak of equivalence classes when a relation is not an equivalence relation.

Equivalence relation does not mean an "equivalent relation". It is a property of a relation, and not a property between different relations.

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Relation $(1)$ is not reflexive, hence not an equivalence relation. It is lacking the pair $(e, e)$.

Relation $(2)$ is an equivalence relation. Now you need to find the equivalence class of $a$: all the elements that are related to $a$.

Relation $(3)$ is reflexive and symmetric, but not transitive. $(c, b),$ and $(b, d) \in R$, but $(c, d) \notin R$. Since it is not transitive, $R$ cannot be an equivalence relation.

Relation $(4)$ is reflexive and transitive, but not summetric: $(a, c) \in R$, but $(c, a) \notin R$.

Relation $(5)$ is an equivalence relation: it is reflexive, symmetric, and transitive. Now you need to find the equivalence class of $a$: [a]. This is the set of all elements in the set which are related to $a$.

Relation $(6)$ is also an equivalence relation: it is reflexive, symmetric, and transitive. Now you need to find the equivalence class of $a$: $[a]$. This is the set of all elements which are related to $a$.

Answer 2

Notice that if 3 is an equivalence relation, then since $<b,d>$ is in the relation, so should $<d,b>$, but it's not there, and therefore, this isn't an equivalence relation.

Answer 3

It looks to me like none of these are equivalence relations. According to the Wikipedia entry, an equivalence relation is reflexive, symmetric, and transitive. Relationship (3) fails to be transitive; (4) is neither transitive nor symmetric. But they all fail to be reflexive, because the element $f$ is nowhere in sight.

Added later: The answer above was to the original version of the question, which included an element $f$. In the edited version, only (1) fails to be reflexive. The equivalence classes for $a$ in the others are as the OP gave them.