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:
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.)