Question

Find the intersection of each pair of set
i.\[X = \left\{ {1,3,5} \right\},Y = \left\{ {1,2,3} \right\}\]
ii.\[A = \left\{ {a,e,i,o,u} \right\},B = \left\{ {a,b,c} \right\}\]
iii.$A = \left\{ {x:x\,is\,a\,natural\,number\,and\,multiple\,of\,3} \right\}$ $B = \left\{ {x:x\,\,is\,a\,natural\,number\,less\,than\,6} \right\}$
iv.$A = \left\{ {x:x\,is\,a\,natural\,number\,and\,1 < x \leqslant 6} \right\}$ $B = \left\{ {x:x\,is\,a\,natural\,number\,and\,6 < x < 10} \right\}$
v.$A = \left\{ {1,2,3} \right\},B = \phi $

Answer 1

Hint: The intersection between two sets can be determined as finding the elements which is common in both of the sets and the collection of the common elements of both the sets is known as intersection of those set that means to find the intersection of the given options we need to find the common elements between the sets.

Complete step-by-step answer:
i.Now, \[X = \left\{ {1,3,5} \right\},Y = \left\{ {1,2,3} \right\}\]
The first step we need to look on the common elements as in the above the element X has is
\[X = \left\{ {1,3,5} \right\}\]
And Y has
\[Y = \left\{ {1,2,3} \right\}\]
So, we can see that the common elements between the two sets are $1\,and\,3$and the collection of the common elements are known as the intersection
$X \cap Y = \left\{ {1,3} \right\}$
Hence, the above is the intersection of X and Y
Now,
ii.\[A = \left\{ {a,e,i,o,u} \right\},B = \left\{ {a,b,c} \right\}\]
The first step we need to look on the common elements as in the above the element A has is
\[A = \left\{ {a,e,i,o,u} \right\}\]
And B has
\[B = \left\{ {a,b,c} \right\}\]
So, we can see that the common elements between the two sets are a and the collection of the common elements are known as the intersection
$A \cap B = \left\{ a \right\}$
Hence, the above is the intersection of A and B
Now,
iii.$A = \left\{ {x:x\,is\,a\,natural\,number\,and\,multiple\,of\,3} \right\}$ $B = \left\{ {x:x\,\,is\,a\,natural\,number\,less\,than\,6} \right\}$

In the given case the intersection can be determined as first determining the elements of set
Let us discuss about A

$A = \left\{ {x:x\,is\,a\,natural\,number\,and\,multiple\,of\,3} \right\}$
The natural numbers, which is multiple of A, we can get it as
$A = \left\{ {3,6,9,12,15....} \right\}$
The above represent the set which involves the elements the natural number which is the multiple of $3$
$B = \left\{ {x:x\,\,is\,a\,natural\,number\,less\,than\,6} \right\}$
Representing the element
$B = \left\{ {1,2,3,4,5} \right\}$
Now, we find the intersection of the element is the collection of the common elements between two sets
$A \cap B = \left\{ {3,6,9,12,15..} \right\} \cap \{ 1,2,3,4,5\} = \left\{ 3 \right\}$

And the next part
iv.$A = \left\{ {x:x\,is\,a\,natural\,number\,and\,1 < x \leqslant 6} \right\}$ $B = \left\{ {x:x\,is\,a\,natural\,number\,and\,6 < x < 10} \right\}$
The natural numbers, which is multiple of A , we can get it as
$A = \left\{ {2,3,4,5,6} \right\}$
The above represent the set which involves the elements the natural number which satisfies $1 < x \leqslant 6$
$B = \left\{ {x:x\,is\,a\,natural\,number\,and\,6 < x < 10} \right\}$
Representing the element
$B = \left\{ {7,8,9} \right\}$
Now, we need to find the intersection between the sets are
$A \cap B = \left\{ {2,3,4,5,6} \right\} \cap \left\{ {7,8,9} \right\}$
Hence, we can write it does not involve any common elements
$A \cap B = \phi $
The above represent the required intersection.

Now, the last part
v $A = \left\{ {1,2,3} \right\},B = \phi $
Finding the common elements, we get
$A = \left\{ {1,2,3} \right\}$
And $B = \phi $
Hence, the common elements between them are
$A \cap B = \left\{ {1,2,3} \right\} \cap \phi $
Hence, it has no common elements and can be represented as
$A \cap B = \phi $
The above is the representation of the intersection between the two sets


Note: The set which is not defined in terms of elements but define in terms of sentences then in that case we need to find the value we need to open the set For example write the elements of the set involves the natural numbers which are multiple of $3$ then in that case we have the value $3,6,9,12,15..\,and\,so\,on$.