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Trignometry
If n (A)=x and n(B)=y,,,,,,then what is the total no. of function from A to B when function are to be--------------------------------------(a) one-one & (b) onto......................please explain both the case separately.............
Comments (6)
if A has x elements and b has y elements...
then clearly for injective mapping...
first element of A can go to any element of B,2nd element of A go to any (y-1) avlues....and so on.
hence there will be.......x*(x-1)*(x-2).....
or
in short there will be yPx
it will be better to go to ONTO with an example...
suppose A={1,2,3,4} and B={1,2}
now , there will be 24 functions from A to B.
now if all d elements of A go to {1}of B..i.e{ 2} is left out hence it is an into..
similarly if all go to 2..1 is left out..
hence there will be ...
16-2...
14 onto functions.....
if any body give general formula...welcome....
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NUMBER OF ONE ONE FUNCTIONS = yPx
number of functions in which range contains exactly one element = yC1.1x
number of functions in which range contains at most one element = yC2.2x
number of functions in which range contains exactly two element = yC2.(2x - 2C1.1x)
number of functions in which range contains at most three element = yC3.3x
number of functions in which range contains exactly three element = yC3.{3x - 3C2.(2x - 2C1.1x) - 3C1.1x}
similary continue for exactly 4, then five and finally upto y elements to get number of onto functions.