[Math] Elementary Set Theory – Relations

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I'm not exactly sure what to search for this problem I'm having, as I don't know the keywords, so I figured the best action would be to ask a question.

I have this question:

"Consider A = {1, 2, 3, 4, 6}. Let R be the relation "x divides y (xly) IF and ONLY IF there exists an integer z such that xz = y.

Write R as a set of ordered pairs."

Firstly, where it says "x divides y (xly), should this be (xly) or (x|y)? I am more inclined to believe it should be (x|y), but i'm not sure.

I'm not exactly sure how I can get the relation. Can someone help me out, and tell me how you did it?

Thanks.

Best Answer

There are $6×6 = 36$ ordered pairs in $A×A$. Just write them down in a structured way and strike out all the pairs which don’t satisfy the relation. And then $R$ is what is left.

And $x \mid y$ means that $x$ divides $y$, e.g $2 \mid 4$ and $2 \mid 6$, but $3 \nmid 4$.

Let me clarify some points: $R$ is a relation, which you can interpret in two ways:

  • formally $R$ is a subset of $A × A$, i.e. $R ⊂ A×A$, and
  • intuitively it’s condition by which you call an element $x ∈ A$ related to some other (not necessarily different) element $y ∈ A$.

The order is important. If $x$ is related to $y$ by condition $R$, one often writes $xRy$.

And in your definition, an element $x ∈ A$ is related to $y ∈ A$ by $R$ if and only if $x$ divides $y$, or more formally $xRy :⇔ x \mid y$ or even more formally $(x,y) ∈ R :⇔ x \mid y$ or maybe even $R := \{(x,y) ∈ A×A;\; x\mid y\}$.

These are all definitions of your relation $R$. You have to figure out, what $R$ looks like, i.e. write $R = \{ (1,6), … \}$. (Since $1 \mid 6$ you have $(1,6) ∈ R$, but what else is in there? Fill out for the dots.)

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