Equivalence relations and classes problem.

[pbxed]

Homework Statement

Let X = {a,b,c,d}. How many different equivalence relations are there on X? What subset of
XxX corresponds to the relation whose equivalence classes are {a,c},{b,d}

Homework Equations

N/A

The Attempt at a Solution

So I wrote out all the possible "blocks" and it comes to 15 which is the bell number B4 = 15 so that's correct.

My problem comes from the second part of the question because I'm unsure what its asking. Is it just asking for the 15 different partitions possible which one is related to {a,c},{b,d} ?

if that is the case isn't it just {{a,c},{b,d}} ~ {a,c,{b,d}} ?

Thanks in advance

 

[mighty2000]

for any help or clarification!

it is important to be clear and precise in your communication. In this case, the question is asking for the number of different equivalence relations on the set X, not the number of partitions. The two concepts are related, but they are not interchangeable.

To answer the question, we first need to understand what an equivalence relation is. An equivalence relation is a relation on a set that satisfies three properties: reflexivity, symmetry, and transitivity. In simpler terms, it is a relation that is reflexive, symmetric, and transitive.

Now, let's look at the second part of the question. It is asking for the subset of XxX that corresponds to the relation whose equivalence classes are {a,c} and {b,d}. In other words, we are looking for a subset of XxX that satisfies these two conditions:

1. For any element x in X, (x,x) is in the subset (reflexivity)
2. If (x,y) is in the subset, then (y,x) is also in the subset (symmetry)

We can see that this subset would correspond to the equivalence relation on X where {a,c} and {b,d} are the only two equivalence classes. This would be a total of 4 elements in the subset: {(a,a),(c,c),(b,b),(d,d)}. So, to answer the question, there is only one equivalence relation on X that has {a,c} and {b,d} as its equivalence classes.

To summarize, the number of equivalence relations on X is 15 and the subset of XxX that corresponds to the relation whose equivalence classes are {a,c} and {b,d} is {(a,a),(c,c),(b,b),(d,d)}.