Relations and Functions

Relations and Functions

Question

In a group of 400 people, 250 can speak Hindi and 200 can speak English. How many people can speak both Hindi and English?

Answer

Let  A = set of people who can speak Hindi.
Let B = set of people who can speak English.
Number of people = 400                                               space space space space space space rightwards double arrow space space space straight n left parenthesis straight A union straight B right parenthesis space equals space 400 space             ...(i)
Number of people who can speak hindi = 250      space space space space space space space space space space space space space space space space space space space rightwards double arrow space space space space straight n left parenthesis straight A right parenthesis space equals space 250                   ....(ii)
Number of people who can speak English = 200    space space space space space space space space space space space space space space space space space rightwards double arrow space space space space space space straight n left parenthesis straight B right parenthesis space equals space 200                  ...(iii)
Number of people who can speak both Hindi and englsih  = space space space space space space space straight n left parenthesis straight A intersection straight B right parenthesis
We  know that   
space space space space space space space space space space space space straight n left parenthesis straight A union straight B right parenthesis space equals space straight n left parenthesis straight A right parenthesis space plus space straight n left parenthesis straight B right parenthesis space minus space straight n left parenthesis straight A intersection straight B right parenthesis

∴    space space space space space space space space space space space space 400 space equals space 250 space plus space 200 space minus space straight n left parenthesis straight A intersection straight B right parenthesis
space space space space space space space space space space space space rightwards double arrow space space space space space space space space space space space space space straight n left parenthesis straight A intersection straight B right parenthesis space equals space 250 plus 200 minus 400 equals 50
Hence, the number of people, who can speak both languages.
space space space space space space space space space space space space space equals space straight n left parenthesis straight A union straight B right parenthesis equals 50

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