# Thread: Equivalence Relations: Symbol Confusion

1. ## Equivalence Relations: Symbol Confusion

"THE" QUESTION...
Let A = Z (all integers)
and let R be the relation defined on A by a R b if and only if a|b.
Is R an equivalance relation?

"MY" QUESTION...
I'm thrown off by the | symbol - is this saying that a is dividable by b? Which would include the following...
{(1,1),(2,1),(2,2),(3,1),(3,3),(4,1),(4,2),(4,4).. .}

Any help would be greatly appreciated!
Doug

2. The relation R on your question is not an equivalence relation, rather, it is a "partial order" relation.

If R be an equivalence relation, it should satisfy following three conditions.

1. Reflexive ( aRa )
2. Symmetry (if aRb, then bRa)
3. Transitive (if aRb and bRc, then aRc)

(aRb means aRb = TRUE)

Your question satisfies condition 1 and 3 but it does not satisfy the condition 2 ( For instance, 3|6 is true, but 6|3 not ).

3. Thank you. I take it by your answer that you agree a|b means a is divisible by b (or at least b divisible by a in your example of 3,6)?

I haven't seen this symbol (pipe on the keyboard) elsewhere in our textbook, and wasn't sure of its meaning.

4. Originally Posted by dstclair
"THE" QUESTION...
Let A = Z (all integers)
and let R be the relation defined on A by a R b if and only if a|b.
Is R an equivalance relation?
This is a really TRICK question!
Does zero divide zero?
If not, can it be reflexive?

5. Originally Posted by Plato
This is a really TRICK question!
Does zero divide zero?
If not, can it be reflexive?
You are right. My mistake
"|" is not reflexive over all integers because 0|0 is not defined.