Thread: 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

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 ).

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.

Thanks again for your quick and thorough answer!

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?

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.