 # Let P be the power set of {a,b,c} Define a function from P to the set of integers as follows: f(A) = |A| . (|A| is the cardinality of A.) Is f injective? Prove or disprove. Is f surjective? Prove or disprove. Deborah Wyatt 2022-07-18 Answered
Let P be the power set of {a,b,c} Define a function from P to the set of integers as follows: $f\left(A\right)=|A|.\left(|A|$ is the cardinality of A.) Is f injective? Prove or disprove. Is f surjective? Prove or disprove.
I'm having a hard time understanding what this question is asking (as I have with most discrete math problems). What does it mean by "Let P be the power set of..."?
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Step 1
The power set of a set is the set of all subsets. So, for example, for the set {a,b,c}, the power set is:
$\left\{\mathrm{\varnothing },\left\{a\right\},\left\{b\right\},\left\{c\right\},\left\{a,b\right\},\left\{a,c\right\},\left\{b,c\right\},\left\{a,b,c\right\}\right\}$. The function f gives the cardinality of a given subset. For example, $f\left(\left\{a,c\right\}\right)=2$, $f\left(\mathrm{\varnothing }\right)=0$, and so on.
Step 2
Then you have to prove whether the function is injective, i.e. if $f\left(A\right)=f\left(B\right)$ for some subsets A and B, does it have to be the case that $A=B$?
And for surjectivity, is it true that for every integer n, there is a subset $A\subseteq \left\{a,b,c\right\}$ such that $|A|=n$?