Define an equivalence relation on the set A={a,b,c,d} such that the equivalence classes are {a,b,c} and {c}
Hello, sorry if the problem is very trivial, but I'm just learning and I have a question. They ask me to have these equivalence classes [x_{1}]={a,b,c} [x_{2}]={c} that is, in my set R I must have R={(x_{1},a),(x_{1},b),(x_{1},c),(x_{2},c)} certain?
So, for it to be equivalent, I must have R=(a,a),(b,b),(c,c)) and do the same for symmetry and transitivity. but in that case I will have more equivalence relations right? [a]={a} for example. Or am I misunderstanding everything? Could you please explain me, thank you very much.
The equivalence classes of an equivalence relation on $A$ will be a partition of $A$. Therefore $\{a,b,c\}$ and $\{c\}$ cannot be the equivalence classes of any equivalence relation on $A$ (since both sets contain $c$).
If you meant to write $\{a,b,c \}$ and $\{d\}$, then this would be a partition of $A$. In which case you could define an equivalence relation $\sim$ on $A$ by $x \sim y$ if they are in the same part of the partition.