Question

For each function below, determine whether or not the function is injective and whether or not the function is surjective. Be sure to justify your answers.
(b) f: P({1,2,3}) + N given by f(A) = |A| (Note: P(S) denotes the power set of a set S.)

Answer 1

Answer:

The function is neither injective nor surjective

Step-by-step explanation:

f: P({1,2,3}) -> N is given by f(A) = |A|

where P(S) denotes the power set of a set S and |A| the number of elements of A

$P(\left \{ 1,2,3 \right \})=\left \{ \varnothing ,\left \{ 1 \right \},\left \{ 2 \right \},\left \{ 3 \right \},\left \{ 1,2 \right \},\left \{ 1,3 \right \},\left \{ 2,3 \right \},\left \{ 1,2,3 \right \} \right \}$

$f(\varnothing )=0$

$f(\left \{ 1 \right \})=f(\left \{ 2 \right \})=f(\left \{ 3 \right \})=1$

$f(\left \{ 1,2 \right \})=f(\left \{ 1,3 \right \})=f(\left \{ 2,3 \right \})=2$

$f(\left \{ 1,2,3 \right \})=3$

The function is not injective, for  

$P(\left \{ 1 \right \})=P(\left \{ 2 \right \})$

but  

$\left \{ 1 \right \}\neq \left \{ 2 \right \}$

The function is not surjective either, because 5 is a natural number which is not the image of any element A under f, i.e., there is no element in A in P({1,2,3}) such that f(A)=5