Question

(a) Let \(F: A \rightarrow B\) be a function. Prove that \(F\) is onto if and only if \(F^{-1}\left(B_{1}\right) \neq B\) for each nonempty subset \(B_{1}\) of \(B\). (b) Let \(F: A \rightarrow B\) be a function. Prove that \(F\) is onto if and only if \(F\left(F^{-1}\left(B_{1}\right)\right)=\) \(B_{1}\) for all \(B_{1} \subseteq B\).

Short answer

(a) \( F \) is onto if \( F^{-1}(B_1) \neq B \) for nonempty \( B_1 \subseteq B \). (b) \( F \) is onto if \( F(F^{-1}(B_1))=B_1 \) for all \( B_1 \subseteq B \).

Step-by-step solution

Understanding Onto Function

A function \( F: A \rightarrow B \) is onto (surjective) if for every element \( b \in B \), there exists at least one element \( a \in A \) such that \( F(a) = b \). In other words, the range of \( F \) is equal to \( B \).

Proving \( F \) is Onto - Part (a) - Necessity

Assume \( F \) is not onto. Then, there exists an element \( b \in B \) such that there is no \( a \in A \) with \( F(a) = b \). Consider the nonempty subset \( B_1 = \{ b \} \). Since there is no element in \( A \) mapped to \( b \), \( F^{-1}(B_1) = \emptyset \). Hence, \( F^{-1}(B_1) eq B \). This shows necessity, if \( F : A \to B \) is not onto, there exists at least one \( B_1 \subseteq B \) such that \( F^{-1}(B_{1}) eq B \).

Proving \( F \) is Onto - Part (a) - Sufficiency

Assume that for every nonempty subset \( B_1 \subseteq B \), \( F^{-1}(B_1) eq B \). Suppose \( F \) is not onto. Then, there exists some \( b \in B \) with no preimage, making \( F^{-1}(\{ b \}) = \emptyset \), contradicting our assumption. Hence, \( F \) must be onto if the inverse condition holds for all nonempty subsets.

Proving \( F \) is Onto - Part (b) - Necessity

Assume \( F \) is onto. Take any subset \( B_1 \subseteq B \). For any \( b \in B_1 \), there exists \( a \in A \) such that \( F(a) = b \) because \( F \) is onto. Thus, \( F(F^{-1}(B_1)) = B_1 \) since all elements of \( B_1 \) have preimages in \( A \). Necessarily, \( F(F^{-1}(B_1)) = B_1 \) for onto \( F \).

Proving \( F \) is Onto - Part (b) - Sufficiency

Assume for all \( B_1 \subseteq B \), \( F(F^{-1}(B_1)) = B_1 \). Consider \( B_1 = B \). Then \( F(F^{-1}(B)) = B \). Therefore, the range of \( F \) covers the entire set \( B \), which means \( F \) is onto. Thus, sufficiency is established showing \( F \) must be onto if \( F(F^{-1}(B_1))=B_1 \) holds for all subsets.

Key concepts

Function Inverse

An inverse function reverses the action of a given function. For a function \( F: A \rightarrow B \), the inverse, denoted \( F^{-1} \), maps elements from \( B \) back to \( A \). This modeling works if each element in \( B \) has a unique preimage in \( A \). However, an inverse does not always exist for every function.

• The inverse exists if the function is bijective (both injective and surjective).
• For a surjective function, each element in \( B \) has at least one preimage, but their uniqueness is not guaranteed.

Understanding the inverse operation enhances our grasp of functions' behaviors, particularly when proving a function's characteristics such as being onto.

Surjective Function

A surjective function, also known as an onto function, is a critical type of function where the range covers the entire codomain \( B \). For \( F: A \rightarrow B \) to be surjective, each element \( b \in B \) must be mapped from at least one element \( a \in A \). This ensures no element in \( B \) is left unmapped.

• If a function is onto, its inverse defined subset mappings will align correctly, satisfying conditions for proofs.
• If a function is not onto, some elements \( b \) in \( B \) remain without preimages, invalidating certain inverse claims.

Recognizing and analyzing surjective functions help determine which function characteristics hold true, aiding in deeper mathematical understanding.

Subset Mapping

Subset mapping involves examining how a function \( F: A \rightarrow B \) maps elements to subsets of \( B \). This practice is crucial when verifying if a function is onto by observing how different subsets \( B_1 \subseteq B \) relate back to \( A \) through the inverse.

• If \( F \) is onto, for every subset \( B_1 \), there must be adequate mapping from \( A \) to cover \( B_1 \).
• For surjective verification, consider \( F(F^{-1}(B_1)) = B_1 \). This must hold across all possible subsets of \( B \).
• Analyzing specific cases, like letting \( B_1 \) equal \( B \), can test the overall mapping capability of the function.

Understanding how functions handle subset mappings enables one to effectively determine if a function meets specific mathematical conditions like being onto.