Question

A relation $R$ defined on the set of all non-negative integers where, $aRb$ if and only if $b=a^k$ for some positive integer $k.$ Then select the corrrect statement(s).

• A. $Codom\left(R\right)=\{0,1,2,3,4,5,...\}$
• B. $Range\left(R\right)=\{a^k:a\in Z,k>0\}$
• C. $R=\left\{\right(a,b)\in W\times W:b=a^k\}$
• D. $Dom\left(R\right)=\{0,1,2,3,4,5,...\}$

Answer 1

Answer: (A), (C), (D)

$Dom\left(R\right)=\{0,1,2,3,4,5,...\}$

The set of all non-negative integers is $\{0,1,2,3,4,5,...\}$

We denote this set by $W.$

The relation $R$ in the set-builder form is,

$R=\left\{\right(a,b)\in W\times W:b=a^k\}$

So, the domain is,

$Dom\left(R\right)=W=\{0,1,2,3,4,5,...\}$

But, for each $a$ in the domain $W$ and for all positive integers k, the image of a under R, which is, $R\left(a\right)=b=a^k$ can take many values which can only be represented in set-builder form.

So, the range of the relation R is,

$Range\left(R\right)=\{a^k:a\in W,k>0\}$

And, the Codomain is,
$W=\{0,1,2,3,4,5,...\}$