Let S = {1, 2} and T = {a, b, c}. (a) How many unique functions are there mappin
ID: 3850313 • Letter: L
Question
Let S = {1, 2} and T = {a, b, c}. (a) How many unique functions are there mapping S rightarrow T? (b) How many unique functions are there mapping T rightarrow S? (c) How many onto (surjective) functions are there mapping S rightarrow T? (d) How many onto (surjective) functions are there mapping T rightarrow S onto functions there are)? (c) How many one-to-one (injective) functions are there mapping S rightarrow T? (f) How many one-to-one (injective) functions are there mapping T rightarrow S? (g) Let f: S rightarrow T, is it possible to define f^-1? Why or why not?Explanation / Answer
a)
for each element from S has exactly one value mapped onto T.
hence for each element from S has 3 options, one of them will be there in function.
There are 2 elements in S. each of them has 3 options.
Hence there are 3^2 = 9 unique functions possible.
b)
similarly,
for each element of T has 2 options and there are 3 elements in T.
hence there are 2^3 = 8 unique functions possible.
c)
here T has 3 elements and S has only 2 elements.
So for any function, at most 2 elements from T are mapped with elements of S.
hence, at least 1 element b will be there in T such that no for all a in S , f(a) = b is not true.
hence there would not be any onto function on S->T.
d)
for any fucntion T->S, if it is onto function then ,
each element of S has its preimage.
here there are 3 elements in T and 2 elements in S.
Two elements from S has choice of 3 elements from T.
hence first element of S hs 3 choices and second element of S has 2 choices.
Number of choices = 3*2 = 6.
e)
Each element from S has 3 choices, but for function to be one-to-one, no two elements can map to same elements from T.
hence number of one-to-one functions =3*2 = 6
f)
There is no such function possible.
as there are 3 elements in T and only 2 elements in S.
For function to be one-to-one , S must have at least 3 elements.
hence no such function possible.
g)
f:S->T is one to one but not onto .
hence inverse of F is not possible.
if you have any doubts then you can ask in comment section.
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