Question

Let \(A\) and \(B\) be nonempty sets, and let \(F: A \rightarrow B\) be a function. Prove that the following are equivalent: (a) \(F\) is onto. (b) There is a function \(G: B \rightarrow A\) such that \(F \circ G=I d_{B}\). (c) For any set \(C\) and for functions \(H_{1}: B \rightarrow C\) and \(H_{2}: B \rightarrow C,\) if \(H_{1} \circ F=H_{2} \circ\) \(F,\) then \(H_{1}=H_{2}\)

Short answer

The function \( F \) being onto is equivalent to both properties: existence of function \( G \) with \( F \circ G = I_d_B \), and if \( H_1 \circ F = H_2 \circ F \), then \( H_1 = H_2 \).

Step-by-step solution

Understand and Define 'F is Onto'

The 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 \( F(a) = b \). This means every element in \( B \) is an image of some element from \( A \).

Establish Equivalence (a) => (b)

Assume that \( F \) is onto. We need to show that there exists a function \( G: B \rightarrow A \) such that \( F \circ G = I_d_B \). Define \( G(b) = a \), where \( F(a) = b \). This is possible as \( F \) is onto. For every \( b \in B \), \( F(G(b)) = F(a) = b \). So, \( F \circ G(b) = b \) for every \( b \in B \), showing \( F \circ G = I_d_B \).

Establish Equivalence (b) => (c)

Assume there exists \( G: B \rightarrow A \) such that \( F \circ G = I_d_B \). We need to prove that if \( H_1 \circ F = H_2 \circ F \), then \( H_1 = H_2 \). For any \( b \in B \), \( H_1(b) = H_1(F(G(b))) = H_2(F(G(b))) = H_2(b) \) because \( F \circ G = I_d_B \). Thus, \( H_1 = H_2 \).

Establish Equivalence (c) => (a)

Assume that for any \( H_1, H_2: B \rightarrow C \), if \( H_1 \circ F = H_2 \circ F \), then \( H_1 = H_2 \). We need to show \( F \) is onto. Consider \( H_1, H_2: B \rightarrow B \) where \( H_1 \) is the identity function and \( H_2 = F(a) = c \) for some fixed \( c \in B \). If \( F \) is not onto, then \( H_1 \circ F eq H_2 \circ F \) for some \( a \in A \), contradicting the assumption. Hence, \( F \) must be onto.

Key concepts

Onto Functions

Onto functions, also known as surjective functions, are an essential concept in discrete mathematics. Consider two sets,

• Set A (the domain)
• Set B (the codomain)

An onto function from \( A \rightarrow B \) points that every element in \( B \) is mapped from one or more elements in \( A \). However, elements in \( A \) may have identical images or some might even remain unmapped. Thus, the core principle is:

• Each element in \( B \) is covered under some mapping from \( A \).

This means no element in \( B \) is left without a pre-image from \( A \). Onto functions are highly significant because they guarantee coverage of all possible outputs, furnishing a level of completeness in practical applications.
When you deal with onto functions, the primary goal is to establish that each target has a valid connection back to its source. This attribute is critical when working with problems involving equivalency proofs, leading to deeper understanding and robust mathematical structures.

Equivalence Proofs

Equivalence proofs are a pivotal aspect of proving logical statements in mathematics. They often illustrate that several different conditions or statements can mean the same thing.
This is executed by proving:

• Statement A is equivalent to Statement B
• Statement B is equivalent to Statement C
• And so on, until a closed loop of equivalence is established

In the context of onto functions, we use equivalence proofs to demonstrate the alignment of three different conditions that characterize onto functions. The aim is to show how the satisfaction of one condition leads to the satisfaction of another condition, thereby proving their interrelations.
Often, we rely on these proofs to break down complex problems into smaller, manageable sub-problems. Substantiating each part helps us piece together a full understanding of any proposition that we might encounter, such as the equivalence process featured in the given exercise.

Surjective Functions

The concept of surjective functions in discrete mathematics is essentially synonymous with onto functions. Understanding surjectivity deepens your comprehension of function mappings.
A function \(F: A \rightarrow B \) is termed surjective if every element in set \( B \) is the image of some element from set \( A \). Similarly put:

• All of \( B \) lines up with some element in \( A \)
• Every target is hit

Surjective functions assure that there are no "loose ends" or elements in \( B \) without corresponding predecessor in \( A \). The exercise relates surjective functions to the existence of a function \( G: B \rightarrow A \) satisfying \( F \circ G = I_d_B \), ensuring each element is neatly matched up.
This helps in analysis and designing systems or mathematical proofs where such thoroughness and guarantees of coverage are required. The study and application of surjective, or onto, functions have use in various practical fields, including coding theory and cryptography.