Question

For sets \(X, Y,\) and \(Z\), let \(F: X \rightarrow Y\) and \(G: Y \rightarrow Z\) be \(I-I\) correspondences. Prove that \((G \circ F)^{-1}=F^{-1} \circ G^{-1}\)

Short answer

The inverse of the composition is the composition of the inverses in reverse order.

Step-by-step solution

Understanding the Problem

We are given two functions: \(F: X \rightarrow Y\) and \(G: Y \rightarrow Z\), both of which are one-to-one and onto, meaning they are bijections. We need to prove that the inverse of the composition of these two functions equals the composition of their inverses in reverse order, i.e., \((G \circ F)^{-1} = F^{-1} \circ G^{-1}\).

Properties of Bijections

Since \(F\) and \(G\) are bijections, they have inverses, \(F^{-1}: Y \rightarrow X\) and \(G^{-1}: Z \rightarrow Y\), respectively. By the definition of inverses, we know that for any element \(x \in X\), \(F^{-1}(F(x)) = x\) and for any \(y \in Y\), \(G^{-1}(G(y)) = y\).

Definition of Composition

Recall that the composition \(G \circ F\) is defined as \((G \circ F)(x) = G(F(x))\) for any \(x \in X\). The objective is to find the inverse of this composite function.

Finding the Inverse of the Composition

The inverse \((G \circ F)^{-1}(z)\) is the function that takes an element \(z \in Z\) and retrieves the original element \(x \in X\). Since \(G(F(x)) = z\), we apply the inverse of \(G\) to recover \(F(x)\), and then apply the inverse of \(F\) to recover \(x\). Thus, \((G \circ F)^{-1}(z) = F^{-1}(G^{-1}(z))\).

Conclude the Proof

From the previous step, we conclude that \((G \circ F)^{-1}(z) = F^{-1}(G^{-1}(z))\), which is equivalent to \((G \circ F)^{-1} = F^{-1} \circ G^{-1}\). This proves that the inverse of the composition is indeed the composition of the inverses in reverse order.

Key concepts

Bijection

A bijection is a special type of function that connects every element of one set to exactly one unique element of another set. It does two things at once. First, it matches each element from the first set with one in the second set without leaving any out. This is called being "onto." Second, each element from the second set is matched with only one from the first set, which is called being "one-to-one."
Bijections are important because they have inverses, meaning we can reverse the function and still have a valid function from the second set back to the first. In our original problem, the functions are bijections, allowing us to discuss and calculate their inverses. When a function is bijective, we can trust that every output has a direct link back to a single input, and vice-versa. This forms a strong foundational understanding of function inverses and is crucial in proving statements about inverse functions.

Function Composition

Function composition involves combining two functions to form a new function. It's like building with blocks, where one function's output becomes another's input. In mathematical terms, for functions \( F: X \rightarrow Y \) and \( G: Y \rightarrow Z \), the composition \( G \circ F \) is the new function defined by \( (G \circ F)(x) = G(F(x)) \) for all elements \( x \in X \).
This composition is a critical operation when working with functions as it allows us to combine transformations represented by each function into a single transformation. Being adept at function composition helps in understanding how complex relationships between sets can be broken down into simpler, manageable steps. In the given problem, the challenge was to find the inverse of such a composition. Understanding that the inverse of a composite function is the composition of each function's inverse—applied in reverse—is key to solving the problem. This fundamental principle often appears in algebra and calculus.

Set Theory

Set theory is the underlying framework used in mathematics to deal with collections of objects, known as sets. Understanding sets and their functions, such as bijections and compositions, hinges on set theory. A set provides a way to group elements and clearly describe relationships between them.
In our context, sets \( X, Y, \) and \( Z \) are collections of elements over which functions and their inverses operate. By examining how functions act between these sets, set theory allows us to see how elements move from one collection to another.

• Set \( X \) may contain all possible inputs of the function \( F \).
• Set \( Y \) is the intermediate set that acts as both output from \( F \) and input to \( G \).
• Set \( Z \) is the set of outputs from the function \( G \).

Set theory provides the vocabulary and rules needed to discuss these collections meaningfully. It forms the bedrock upon which more advanced mathematical concepts and proofs, like the inverse of composite functions, are built. By understanding sets and their operations, students can better grasp the movement and transformation of elements within mathematical systems.