Chapter 12, Problem 12GR

To determine

To find: the union of the two set $X$ and $Y$ .

Answer to Problem 12GR

  $n\left(X\cup Y\right)=\left\{1,2,3,5,6,7,9,10,11,13,15,16\right\}$ .

Explanation of Solution

Given:

  $\begin{array}{l}X=\left\{x|x\text{is an odd whole number less than }16\right\}\\ Y=\left\{2,6,9,10,16\right\}.\\ Z=\left\{z|z\text{is an even whole number less than }19\right\}\end{array}$

Concept used:

In a set theory, the union of a collection of sets is the set of all elements in the collection.

It is one of the fundamental operations through which sets can be combined and related to each other.

According to the law of sets function and Venn diagram.

  $n\left(X\cup Y\right)=n\left(X\right)+n\left(Y\right)-n\left(X\cap Y\right)$ .

Calculation:

According to the given:

  $\begin{array}{l}X=\left\{x|x\text{ is an odd whole number less than }16\right\}\\ Y=\left\{2,6,9,10,16\right\}.\\ Z=\left\{z|z\text{ is an even whole number less than }19\right\}\end{array}$

According to the law of sets function and Venn diagram.

  $\begin{array}{l}n\left(X\cup Y\right)=n\left(X\right)+n\left(Y\right)-n\left(X\cap Y\right)\\ \Rightarrow n\left(X\cup Y\right)=n\left\{1,3,5,7,9,11,13,15\right\}+n\left\{2,6,9,10,16\right\}-n\left\{9\right\}.\\ \Rightarrow n\left(X\cup Y\right)=\left\{1,2,3,5,6,7,9,10,11,13,15,16\right\}.\\ \left\{1,2,3,5,6,7,9,10,11,13,15,16\right\}.\end{array}$

Hence, $n\left(X\cup Y\right)=\left\{1,2,3,5,6,7,9,10,11,13,15,16\right\}$ .