how many unique functions?

Question

I am not sure how to answer this question
Let S = {1, 2} and T = {a, b, c}.

How many unique functions are there mapping S → T? (Is it 2? because (1,a) and (2,b) )

How many onto (surjective) functions are there mapping T → S

How many one-to-one (injective) functions are there mapping S → T?

Let f : S → T, is it possible to define f^(−1) <-- this is the inverse function
? Why or why not?
My answer is yes because since the S elements mapped exactly to one element of the set T

Thanks,

Answer 1

(1) You need to remember the definition of function. f:S -> T is a function if f assigns exactly one element in T for every element in S. Your example of {(1,a), (2,b)} is only one such function. Since you have two elements in S how many ways may they be assigned to elements in B? {(1,a),(2,a)} is another such function.

 

Can you figure out the rest of this one?

 

 

(2) For surjections from T -> S remember all of S must be used e.g. {(a,1),(b,2),(c,1)} works.

 

How many more are there?

 

 

(3)  For injections from S  -> T two elements in S cannot go to the same element in T; e.g {(1,a),{2.b)} works.

 

Again how many more are there?

 

 

(4) Finally for f from S to T does there exist an inverse function. For f^-1 to exist then f  must be both and injection and a surjection. Can you find such an f?

 

 

Think about these and let me know if you need more help. If so on which part(s).