# Equivalence Relation ~

let S = {1,2,3,4}

Explain why each of the below is not an equivalence relation.

{ (1,1), (1,2), (2,1), (2,2), (3,3) }

{ (1,1), (1,2), (2,3), (1,3), (2,2), (3,3), (4,4) }

{ (1,1), (2,2), (3,3), (4,4), (2,3), (3,2), (2,4), (4,2)}

I am having difficulty trying to understand the 3 condition

1. Reflexive
2. Symmetric
3. Transitive

Would appreciate if anyone would to provide me with the guidance.

Thanks

• so do you mean that for 2nd set, (1,2) & (2,1) is not within the set and hence not symmetric? May 11, 2014 at 8:50

1. The first one is surely not an equivalence relation as it is not reflective: $(4,4)$ does not belong, this is $4$~$4$ is false, this is $x$~$x$ for all $x$ is false.
2. The second one is exactly as you wrote in your comment: $(1,2)$ belongs to the relation, $(2,1)$ does not, hence symmetric property fails.
3. Third: we have the pairs $(3,2)$ and $(2,4)$, but not $(3,4)$. Hence is not true that
$x$~$y\ \&\ y$~$z\ \Rightarrow\ x$~$z\quad \forall\,x,y,z$ (not transitive).