Minimal elements

Question

Let $A$ be the set $A = \{1,2,3,...,20\}$. $R$ is the relation over $A$ such that $xRy$ iff $y/x = 2^i$, $i$ is natural including $0$. I am supposed to find the minimal and maximal elements in relation to $R$. Does that mean the elements are members of $A$ or members of $R$? The elements of $R$ are pairs that their divisions yield $1,2,4,8,16 \ldots$ But I am confused as to whether I am searching for pairs in $R$ or numbers in $A$. Thanks.

Answer 1

You are searching for numbers in $A$. This relation is actually a partial order on $A$.

The minimal elements are all the elements $x\in A$ such that there is no $y\neq x$ with $yRx$ (nothing "smaller").

The maximal elements are all the elements $x\in A$ such that there is no $y\neq x$ with $xRy$ (nothing "larger").

In both cases there may be more than one such element. For example, if you only took the set $\{1,2,3,4\}$ and the same relation, we have $1R2$, $2R4$, $1R4$ (and every element is related to itself). This means the minimum elements are $1$ and $3$, whereas the maximal elements are $4$ and $3$.

Answer 2

The relation $R$ can be shown to be a partial order. You must find the least and greatest elements of $A$ according to this ordering relation.

Answer 3

You're searching for members of $A$.

$a \in A$ is maximal, whenever $aRa'$ implies $a'=a$.

$a \in A$ is minimal, whenever $a'Ra$ implies $a'=a$.