Question

Prove that if \(\Phi\) is a function from a set \(\mathrm{A}\) to a set \(\mathrm{B}\) then \(\Phi\) has an inverse if and only if \(\Phi\) is bijective.

Short answer

In summary, we proved that if \(\Phi\) has an inverse, then \(\Phi\) is bijective by demonstrating that \(\Phi\) is both injective and surjective. Conversely, we showed that if \(\Phi\) is bijective, then \(\Phi\) has an inverse by constructing an inverse function \(\Phi^{-1}\) and proving that it satisfies the properties of an inverse function.

Step-by-step solution

Proof 1: If \(\Phi\) has an inverse, then \(\Phi\) is bijective

To prove this, we need to show that \(\Phi\) is both injective (one-to-one) and surjective (onto).

Proving injectivity (one-to-one)

Suppose \(\Phi\) has an inverse, which we will call \(\Phi^{-1}\). By the definition of an inverse function, we have: 1. \(\Phi(\Phi^{-1}(y)) = y\) for all \(y \in B\) 2. \(\Phi^{-1}(\Phi(x)) = x\) for all \(x \in A\) Now, let's suppose there are two elements \(x_1\) and \(x_2\) in set \(A\) such that \(\Phi(x_1) = \Phi(x_2)\). Apply \(\Phi^{-1}\) to both sides: \(\Phi^{-1}(\Phi(x_1)) = \Phi^{-1}(\Phi(x_2))\) Using property (2) above, we get: \(x_1 = x_2\) Thus, \(\Phi\) is injective.

Proving surjectivity (onto)

As per the definition of an inverse function, we know that: \(\Phi(\Phi^{-1}(y)) = y\) for all \(y \in B\) This means that for any element \(y\) in the set \(B\), there exists an element \(x\) in the set \(A\) such that: \(x = \Phi^{-1}(y)\), and \(\Phi(x) = y\) Hence, \(\Phi\) is surjective. By proving both injectivity and surjectivity, we have shown that if \(\Phi\) has an inverse, then it is bijective.

Proof 2: If \(\Phi\) is bijective, then \(\Phi\) has an inverse

Now we need to show that if \(\Phi\) is bijective, then it has an inverse. If \(\Phi\) is bijective, it means it is both injective and surjective.

Defining a candidate inverse function

Let's define a function \(\Phi^{-1}: B \rightarrow A\) such that: \(\Phi^{-1}(y) = x\) if and only if \(\Phi(x) = y\) Since \(\Phi\) is bijective, this function is well-defined for all \(y \in B\).

Proving the inverse function properties

Now we need to prove that \(\Phi^{-1}\) satisfies the properties of an inverse function: 1. \(\Phi(\Phi^{-1}(y)) = y\) for all \(y \in B\) 2. \(\Phi^{-1}(\Phi(x)) = x\) for all \(x \in A\)

Step 2a: Proving property 1

Let \(y \in B\). Since \(\Phi\) is surjective, there exists an \(x \in A\) such that: \(\Phi(x) = y\) By the definition of \(\Phi^{-1}\), we get: \(\Phi(\Phi^{-1}(y)) = \Phi(x) = y\) Therefore, property 1 holds.

Step 2b: Proving property 2

Let \(x \in A\). Since \(\Phi\) is injective, there is a unique \(y \in B\) such that: \(\Phi(x) = y\) By the definition of \(\Phi^{-1}\), we get: \(\Phi^{-1}(\Phi(x)) = \Phi^{-1}(y) = x\) Therefore, property 2 holds. By proving both properties, we have shown that if \(\Phi\) is bijective, then it has an inverse. This concludes the proof that a function \(\Phi\) has an inverse if and only if it is bijective.

Key concepts

Injective Function

An injective function, also known as a one-to-one function, is a type of function where each element of the function's domain (the set A) is mapped to a unique element in its codomain (the set B). This means no two different elements in A are connected to the same element in B.

To visualize this, imagine that each student in a classroom (set A) has a unique locker (set B). No two students share the same locker, so each person's belongings are distinct. In mathematical terms, if a function \( \Phi \) is injective and \( \Phi(x_1) = \Phi(x_2) \), then it must be that \( x_1 = x_2 \).

An easy way to test for injectivity is by using the Horizontal Line Test. If every horizontal line intersects the graph of the function at most once, then the function is injective. Injectivity is crucial for a function to have an inverse because it ensures that each output was produced from a distinct input.

Surjective Function

A surjective function, or onto function, occurs when the range of the function (all actual outputs from set B) coincides exactly with its codomain. This means for every element in B, there is at least one element in A that maps to it.

Continuing the classroom analogy, imagine a set of lockers (set B) representing the codomain. The function is surjective if every locker has at least one student's belongings (from set A) inside it. In terms of an equation, if \( \Phi \) is surjective, for every \( y \) in B, there exists at least one \( x \) in A such that \( \Phi(x) = y \).

The Horizontal Line Test can again be useful here; a function is surjective if every horizontal line that you draw at any 'y-value' height will intersect the graph of the function. In the context of an inverse function, surjectivity ensures that all possible outputs from B connect back to at least one input from A, making the process reversible.

Inverse Function

An inverse function essentially reverses the operation of a function. If \( \Phi \) is a function with an inverse, then \( \Phi^{-1} \) is that inverse function, swapping the roles of the domain and codomain. For the inverse to exist, every operation must be reversible; put simply, every output in B can trace back to one and only one input in A.

For the inverse function to work, \( \Phi \) must be both injective and surjective, hence bijective. If we consider a function as a process that assigns students to lockers, the inverse function is the process of going from locker back to student. Alternatively, if \( \Phi \) takes you from city A to city B, then \( \Phi^{-1} \) is the route that takes you back from city B to city A.

To determine the inverse of a function, you switch the 'x' and 'y' in the function's equation and solve for 'y' again. The graph of an inverse function is a reflection of the original function's graph across the line \( y = x \)—meaning if the original function (\forall x) and inverse function (\forall y) were plotted on the same graph, they would mirror each other across this line.