Is f one-to-one? Is f onto? Prove or give a counterexample.

Question

Define f(x) : P({a,b,c}) →N as follows:

for all A ∈P({a,b,c}), f(A) = the number of elements in A.

( P is a power set.)

Is f one-to-one? Prove or give a counterexample.

Is f onto? Prove or give a counterexample.

Answer 1

The function f is not one to one since the two sets {a,b} and {b,c} are not equal but f({a,b})=f({b,c})=2. The function is not onto the set of natural numbers since for example there is no subset of the set {a,b,c} of more than 3 elements. But of course the function is onto its image, which is the set {0,1,2,3}. Every function is onto its image by the definition of the image set.