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(a) Express the above information by Venn diagram along with brief description.

(b) Find the numbers who have passed only in Bangla and English.
(c) Find the union of two sets of the prime factors of the numbers who have
passed and failed in both subjects.
Relation

We know, the capital of Bangladesh is Dhaka and that of India is Delhi and Pakistan
is Islamabad. Here, there is a relation of capital with the country. The relation is
country-capital relations. The above relation can be shown in set as follows:

Relation

Country Capital

Bangladesh Dhaka

India Delhi

Pakistan Islamabad

That is, country-capital relation = {(Bangla desh, Dhaka), (India, Delhi), (Pakistan,
Islamabad)}.
If A and B are two sets, the non-zero subset of R of the Cartesian product AuB of
the sets is called relation of B from A.
Here,R is a subset of AuB set, that is, RŽAuB.

Example 15. Suppose, A { 3 , 5 } and B { 2 , 4 }

?AuB { 3 , 5 }u{ 2 , 4 } {( 3 , 2 ),( 3 , 4 ),( 5 , 2 ),( 5 , 4 )}

?R {( 3 , 2 ),( 3 , 4 ),( 5 , 2 ),( 5 , 4 )}

If the condition is x!y,R {( 3 , 2 ),( 5 , 2 ),( 5 , 4 )}

and if the condition isxy,R { 3 , 4 }

If an element of set A is x and that of the set B is y and (x,y)R, we write
xRy and read as xis related to y. That the element x is R related to element y.
Again, if the relation of a set, from set A that is RŽAuA,R is called A related.
So, if the relation is given between set A and B, non-zero subset of ordered pair
(x,y) with yB related to xA, is a relation.

Example 16. If P { 2 , 3 , 4 },Q { 4 , 6 } and y 2 x is relation under consideration
between the elements of P and Q, find the relation.

Solution : Given that, P { 2 , 3 , 4 } and Q { 4 , 6 }

According to the question, R {(x,y):xP, yQ and y 2 x}

Here,PuQ { 2 , 3 , 4 }u{ 4 , 6 } {( 2 , 4 ),( 2 , 6 ),( 3 , 4 )( 3 , 6 ),( 4 , 4 ),( 4 , 6 )}

?R {( 2 , 4 ),( 3 , 6 )}