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 ( A ) = | A | . ( | A | is the cardinality of A.) Is f injective? Prove or disprove. Is f surjective? Prove or disprove.
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Answers (1)

Cael Cox
Answered 2022-07-19 Author has 11 answers
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:
{ , { a } , { b } , { c } , { a , b } , { a , c } , { b , c } , { a , b , c } }. The function f gives the cardinality of a given subset. For example, f ( { a , c } ) = 2, f ( ) = 0, and so on.
Step 2
Then you have to prove whether the function is injective, i.e. if f ( A ) = f ( B ) 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 { a , b , c } such that | A | = n?
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