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For a set $A$ define $\mathcal{P}(A)$ to be the set of all subsets of $A$. For example, if $A = \{1, 2\}$ then $\mathcal{P} (A) = \{ \emptyset, \{1, 2\}, \{1\}, \{ 2 \} \}$. Let $A \rightarrow \mathcal{P}(A)$ be a function and $A$ is not empty. Which of the following must be TRUE?

  1. $f$ cannot be one-to-one (injective)
  2. $f$ cannot be onto (surjective)
  3. $f$ is both one-to-one and onto (bijective)
  4. there is no such $f$ possible
  5. if such a function $f$ exists, then $A$ is infinite
asked in Set Theory & Algebra by Veteran (79k points)   | 136 views

1 Answer

+9 votes
Best answer

Even if it can be one-to-one in the following way,

But, It cannot be onto,because, the number of elements in domain $(A)$  $<$ the number of elements in co-domain ($P(A)$) . For a function to be onto, the domain should be able to cover all elements of co-domain with each element of the domain having exactly one image in co-domain.
so option(B)

answered by Veteran (11.5k points)  
edited by

Nice solution.Thanks :)



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