Relations and Functions

Relations and Functions

Question

In a committee, 50 people speak French, 20 speak Spanish and 10 speak both Spanish and French. How many speak at least one of these two languages?

Answer

Let A = set of people who can speak  French.
rightwards double arrow n(A)  = 50                  
                                                                                                                               ....(i)
B = set of people who speak Spanish

rightwards double arrow space n(B)  = 20                                                                                                         ....(ii)
Number of people who speak both languages = space space space space space straight n left parenthesis straight A intersection straight B right parenthesis space equals space 10                                ....(iii)
Number of people who speak at least one of the two languages space space space space space space space space space equals straight n left parenthesis straight A intersection straight B right parenthesis            ....(iv)
We know that 
space space space space space space space 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 straight n left parenthesis straight B right parenthesis space minus straight n left parenthesis straight A intersection straight B right parenthesis

∴     space 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 50 space plus 20 space minus 10 space equals space 60
 Hence, the number of people who speak at least one of the two languages
space 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 space equals 60



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