Chapter 3.8, Problem 22PPE

To determine

To calculate: the union and intersection of the given sets and draw a Venn diagram to represent it.

Answer to Problem 22PPE

The Venn diagram that represents the intersection and union of the given sets is below:

  

Explanation of Solution

Given information:

The letter words are: ALGEBRA, GEOMETRY and CALCULUS.

Represented by the sets as:

  $V=\{A,L,G,E,B,R\}$ , $W=\{\text{G},\text{E},\text{O},\text{M},\text{T},\text{R},\text{Y}\}$ and $X=\{\text{C},\text{A},\text{L},\text{U},\text{S}\}$

Calculation:

The intersection of two sets $A$ and $B$ is the set that contains all elements of $A$ that also belong to $B$ but no other elements. In other words, an element is in the intersection if it belongs to all of the sets. That is,

  $A\cap B=\{x\mid x\in A$

  $x\in B\}$

The union of two sets $A$ and $B$ is the collection of points which are in $A$ or in $B$ or in both $A$ and $B$

In other words, an element is in the union if it belongs to at least one of the sets. That is,

  $A\cup B=\{x\mid x\in A$

  $x\in B\}$

The given sets are below:

  $V=\{A,L,G,E,B,R\}$

  $W=\{\text{G},\text{E},\text{O},\text{M},\text{T},\text{R},\text{Y}\}$

  $X=\{\text{C},\text{A},\text{L},\text{U},\text{S}\}$

There are no common elements of the sets $V,W$ and $X,$ which means the intersection of three sets, is a null set.

The common elements of the sets $V$ and $W$ are $G,E$ and $R$ .

The common elements of the sets $V$ and $X$ are $A$ and $L.$

There are no common elements of the sets $W$ and $X$ .

Hence, the Venn diagram that represents the intersection and union of the given sets is below: