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Understanding when two functions are equal is a fundamental concept in mathematics. Two functions, say \( f \) and \( g \), are considered equal if they exhibit three critical characteristics:

• They have identical domains.
• They share the same codomain.
• For every element \( x \) within their common domain, the outputs are exactly the same, i.e., \( f(x) = g(x) \).

This means if you input any element from their domain into both functions, they will yield the same result.
Let's take an example to make this clearer. Consider two functions \( f \) and \( g \), both defined from \( A \) to \( B \). If for every \( a \in A \), \( f(a) \) gives \( g(a) \), then you can say that \( f = g \).
This concept is particularly important when dealing with compositions and identity functions, as these require functions to operate within defined specifics of equality.

Bijections are a special type of function with some unique characteristics. A function \( f: A \rightarrow B \) is a bijection if it is both injective and surjective.

• Injective: Each element in the codomain \( B \) is mapped by at most one element from domain \( A \). Essentially, no two elements in \( A \) map to the same element in \( B \).
• Surjective: Every element in the codomain \( B \) is the image of some element from the domain \( A \). There are no leftovers in \( B \), ensuring full coverage.

Think of bijections as perfect matches or pairings between sets where every element has a unique partner, and there are no unmatched elements left in either set. In the exercise context, having bijections \( f: A \rightarrow B \) and \( g: B \rightarrow A \) implies both functions are inverses of each other.
Therefore, composing them, as in \( g \circ f \) or \( f \circ g \), should result in identity functions, reflecting their uniqueness and complete nature. This property is why bijections are known as "one-to-one" and "onto" functions.

Composition of functions is an essential operation, especially when working with multiple functions. If you have two functions \( f: A \rightarrow B \) and \( g: B \rightarrow C \), their composition \( g \circ f \) is a new function from \( A \) to \( C \).
To compose, you do the following:

• First, apply \( f \) to an element from set \( A \).
• Then, take the result and apply \( g \) to it to get the final output in set \( C \).

It's like chaining them together, where the output of \( f \) becomes the input for \( g \).
For the identity and bijective functions discussed before, compositions like \( g \circ f = I_A \) or \( f \circ g = I_B \) simplify because they map back to the original set perfectly, illustrating one key ability of bijections.
This means applying the composition leads perfectly back to where you started, mimicking the role of an identity function where nothing changes, showcasing a harmony between the function operations.