[Math] Determining if a power set is one to one or onto.

elementary-set-theoryfunctions

Let $P$ be the power set of $\{a,b,c\}$. A function $f: P \to \mathbb{Z}$; the set of integers, follows: For $A$ in $P$, $f(A)=$the number of elements in $A$.

  1. Is $f$ one-to-one? Explain.

  2. Is $f$ onto? Explain.

I'm not sure how to picture this function. I understand how to expand the power set of $\{a,b,c\}$ to get $\{a,b,c\}=\{∅,a,b,c,(a,b),(b,c),(c,a),(a,b,c)\}$ and I also understand what a one-to-one function is and an onto function is, but I'm confused as to what the function the problem is trying to elude to and the wording of the problem seems odd to me.

Any help would be greatly appreciated. Thank you.

Best Answer

Hey guys thanks for the help, I think I figured out the answer thanks to you guys.

If I'm understanding correctly the power set of {abc} will map to an integer based on the cardinality of each of the subsets thus

{null set} maps to 0, {a} maps to 1, {b} maps to 1, (a,b} maps to 2, {a,b,c} maps to 3 and so forth.

Thus we have more than one x value mapping the same y value,{a} and {b} both map to 1 so it is not a one to one function.

Furthermore, if this function is to map to the set of all integers, all of the integers are not covered as we only use 0,1,2 and 3. Therefore it is not onto either.

Hope that helps anyone who needs assistance on this question in the future.

Let me know if I made any mistakes.

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