Chapter 3.8, Problem 36PPE

To determine

To find: the intersection of two sets $M$ and $N$ with the help of given information.

Answer to Problem 36PPE

The intersection of two sets $M$ and $N$ is: $M\cap N=\left\{x\mid x\text{ is a multiple of}15\right\}$ .

Explanation of Solution

Given information:

  $M=\left\{x\mid x\text{ is a multiple of 3}\right\}$

  $N=\left\{x\mid x\text{ is a multiple of 5}\right\}$

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,

  $A\cap B=\{x\mid x\in A\text{and}x\in B\}$

For the given set $N$ and $P$ ,

  $\begin{array}{l}M=\{3,6,9,12,15,18,\dots \}\\ N=\{5,10,15,20,\dots \}\\ M\cap N=\{3,6,9,12,15,18,\dots \}\cap \{5,10,15,20,\dots \}\\ =\{15,30,45,\dots \}\end{array}$

Hence, the intersection of two sets $M$ and $N$ denotes all multiples of $15.$

Thus, $M\cap N=\left\{x\mid x\text{ is a multiple of}15\right\}$