Chapter 3.8, Problem 12P

Find each union or intersection. Let $A=\{1,3,5,7,9\},B=\{x\mid x$ is a positive. odd integer less than $10\},C=\{1,2,4,7\},$ and $D=\{x\mid x$ is a negative integer between $-5\text{ and }-1\}$.

$A\cap C\cap D$

To determine

To find: the set $A\cap C\cap D$ .

Answer to Problem 12P

  $A\cap C\cap D=\varphi$

Explanation of Solution

Given information:

Three sets are given as

  $\begin{array}{l}A=\left\{1,3,5,7,9\right\},B=\left\{x|x\text{is}\text{a}\text{positive}\text{odd}\text{integer}\text{less}\text{than}10\right\}\\ C=\left\{1,2,4,7\right\},D=\left\{x|x\text{is}\text{a}\text{negative}\text{integer}\text{between}-5\text{and}-1\right\}\end{array}$

Formula used:

For any two or more sets, union of the sets gives the elements which are in the either one of the sets and the all elements of all sets. Common elements are considered once. Union is represented by the symbol $\cup$ .

For any two or more sets, intersection of the sets gives the elements which are the only common elements of the sets. Intersection is represented by the symbol $\cap$ .

Calculation:

Consider the given sets.

  $A=\left\{1,3,5,7,9\right\}$

and

Now, the set C can be written as

  $C=\left\{1,2,4,7\right\}$

Now, the set D can be written as

  $D=\left\{x|x\text{is}\text{a}\text{negative}\text{integer}\text{between}-5\text{and}-1\right\}$

Now, $A\cap C\cap D$ contains common elements of A , C and D .

  $\begin{array}{c}A\cap C\cap D=\left\{1,3,5,7,9\right\}\cap \left\{1,2,4,7\right\}\cap \left\{-4,-3,-2\right\}\\ =\varphi \end{array}$