1. ## Help with functions.

Hello,

My teacher has given me a problem to try:
Let A={1, 2, 3, 4}. Define a fcn f: Powerset(A) --> N U {0} by f(x) = |x| (cardinality of x) for x element of Powerset(A).

I need to find:
a) f({∅, {1,2}, {2,3}})
b) ƒ−1({3}).

Now, for a, im pretty sure it's simply f({∅, {1,2}, {2,3}}) = 2, since there is nor curly brackets around ∅, so it doesnt count as an element, and other two are elements of powerset (A).

Regarding b, im not too sure what would be the inverse of the cardinality.

For a, is my logic makes sense or i am completely off?
For b, im not sure how to find the inverse.

Thank you for help in advance

2. ## Re: Help with functions.

Originally Posted by Lowoctave
Let A={1, 2, 3, 4}. Define a fcn f: Powerset(A) --> N U {0} by f(x) = |x| (cardinality of x) for x element of Powerset(A).
I need to find:
a) f({∅, {1,2}, {2,3}})
b) ƒ−1({3}).
$\displaystyle f(\{\emptyset, \{1,2\}, \{2,3\}\})=3$

3. ## Re: Help with functions.

I don't know if it's really an "inverse" since the function is not an injection, but you could treat it as a multi-valued inverse of sorts. In that case it would be all the things that give you the result "3" by that function you described.

Plato's right about the cardinality of a. Remember that ∅ is the same as {}. The empty set is included (subset of) in every set, but not contained (member of) in every set. When the empty set is a member of another set, it's just like any other member. (Note you separate it with a comma just like the others.)

4. ## Re: Help with functions.

So for b) it would be ƒ−1({3}) = {{1, 2, 3}, {1, 2, 4} {1, 3, 4}, {2, 3, 4}}?

5. ## Re: Help with functions.

Originally Posted by Lowoctave
So for b) it would be ƒ−1({3}) = {{1, 2, 3}, {1, 2, 4} {1, 3, 4}, {2, 3, 4}}?
The notation $\displaystyle f^{-1}(\{3\})$ is meaningless.
The images of $\displaystyle f$ are numbers not sets.

You could have $\displaystyle f^{-1}(3)=\{\{1,2,3\},\{1,2,4\}\{1,3,4\},\{2,3,4\}\}$.

6. ## Re: Help with functions.

Thank you very much! Very appreciated