# Find the union of each of the following pairs of sets:

(i) $X = \{ 1,3,5\} Y = \{ 1,2,3\} $

(ii)$A = \{ a,e,i,o,u\} ,B = \{ a,b,c\} $

(iii)$A = \{ x:x$ is a natural number and multiple of $3\} $

$B = \{ x:x$is a natural number less than $6\} $

(iv)$A = \{ x:x$ is a natural number and $1 < x \leqslant 6\} $

$B = \{ x:x$is a natural number and $6 < x < 10\} $

(v) $A = \{ 1,2,3\} ,B = \phi $

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**Hint:**Make use of the definition of union of sets here which says union of two given sets is the smallest set which contains all the elements of both the sets and denoted by $ \cup $ and them start to solve the problem

**Complete step by step solution:**

(i) the given sets are:

$

X = \{ 1,3,5\} \\

Y = \{ 1,2,3\} \\

$

Now, we will find union of $X,Y$, we have

$X \cup Y = \{ 1,2,3,5\} $

(ii) Sets are:

$A = \{ a,e,i,o,u\} $

$B = \{ a,b,c\} $

Now, we will find the value of union $A,B$ we have

$A \cup B = \{ a,b,c,e,i,o,u\} $

(iii) sets are: $A = \{ x:x$is a natural number and multiple of $3\} $

$B = \{ x:x$ is a natural number less than $6\} $

Now, Natural numbers $ = 1,2,3,4,5,6,7,8,9,10$…..

And multiple of $3 = 3,6,9,12,15,....$

So, $A = \{ 3,6,9\} $and $B = \{ 1,2,3,4,5\} $

Now, we will find the union of $A\,and\,B$ set, so

$A \cup B = \{ 1,2,3,4,5,6,9\} $

$A \cup B = \{ x:x$ is a natural number, $x = 1,2,3,4,5$ or a multiple of $3\} $

(iv) $A = \{ x:x$ is a natural number and $1 < x \leqslant 6\} $

$B = \{ x:x$ is a natural number and $6 < x < 10\} $

So, $A = \{ 2,3,4,5,6\} $

$B = \{ 7,8,9\} $

Now, $A \cup B = \{ 2,3,4,5,6,7,8,9\} $

$\therefore A \cup B = \{ x:x \in N$ and $1 < x < 10\} $

(v) $A = \{ 1,2,3\} $

$B = \phi \,or\,\{ \,\} \,$

So, $A \cup B = \{ 1,2,3\} $

**Note:**Students must know about natural numbers, whole numbers, integers, etc.

Natural numbers$ = \{ 1,2,3,4,5,......\} $

Whole numbers $ = \{ 0,1,2,3,4,....\} $

Integers numbers $ = \{ - 4, - 3, - 2, - 1,0,1,2,3,.....\} $

Find the union of sets in accordance to the type of number which the set contains(ie whether natural,real or whole numbers)