So I've been asked to find the # of equivalence relations ~ on the set B = {1,2,3,4,.....,m} for some m where if a|b, then a~b. For some reason I just can't seem to get it... how would one show this with only one sided implication?
There's another part of the question asking how many equivalence relations ~ on B exist such that a~b iff a|b, which is clearly the equivalence relation = (since we're dealing with natural numbers).
Am I missing something/not considering something?
For the first, you are given a bunch of pairs that are part of an equivalence relation and asked how many ways there are to extend the list to make an equivalence relation. The if does not tell you anything about whether a par $(a,b)$ is in the relation if $a \not | b$. Note that $1$ divides every number, so every pair $(1,b)$ is part of your relation. Then by symmetry and transitivity, every pair is part of your relation, so there is only one equivalence relation that satisfies the condition, the relation where $a\sim b$ for all $a,b$.
For the second, we have $(1,2)$ is in $\sim$ and $(2,1)$ is not because of the iff. This is not symmetric, so it is not an equivalence relation.