Question

Let \(E\) be a set, and let \(\mathrm{f}, \mathrm{g}\) be mappings of \(\mathrm{E} \rightarrow \mathrm{E}\), Prove: if \(\mathrm{f}\) and \(g\) are each one-to-one, then the composite mapping fog is one-to-one. (f og is defined by the formula \((f \circ g)(x)=f(g(x)))\)

Short answer

To prove that the composite function fog is one-to-one, we assume (fog)(x₁) = (fog)(x₂) for arbitrary elements x₁ and x₂ in the domain of fog. This implies that f(g(x₁)) = f(g(x₂)). Since f is one-to-one, this implies that g(x₁) = g(x₂). Since g is one-to-one, this implies that x₁ = x₂. By the definition of one-to-one, we can conclude that the composite function (fog) is one-to-one.

Step-by-step solution

Let (fog)(x₁) = (fog)(x₂)

Assume (fog)(x₁) = (fog)(x₂) for arbitrary elements x₁ and x₂ in the domain of fog. This implies that f(g(x₁)) = f(g(x₂)).

Use the definition of one-to-one for function f

Since f is one-to-one, if f(g(x₁)) = f(g(x₂)), then this implies that g(x₁) = g(x₂).

Use the definition of one-to-one for function g

Since g is one-to-one, if g(x₁) = g(x₂), then this implies that x₁ = x₂.

Conclude that (fog) is one-to-one

We have shown that if (fog)(x₁) = (fog)(x₂), then x₁ = x₂. By the definition of one-to-one, we can conclude that the composite function (fog) is one-to-one.

Key concepts

Understanding One-to-One Functions

A function is considered one-to-one, or injective, if it matches each element of its domain with a unique element of its range. This means no two different input values from the domain produce the same output value. In simpler terms, if \( f(x_1) = f(x_2) \), then it must be the case that \( x_1 = x_2 \). Let's delve deeper:

• Being one-to-one ensures that the function has an inverse that is also a function. This quality is crucial for many mathematical applications.
• To determine if a function is one-to-one, you can use various methods, such as the horizontal line test when considering graphical representations.
• This uniqueness characteristic greatly affects composition operations, as we'll see with composite functions.

Understanding this concept helps in determining whether certain compositions retain the properties of the original mappings.

Function Composition in Focus

Function composition involves taking two functions and creating a new function by applying one function to the results of another. It's written as \( (f \circ g)(x) = f(g(x)) \). The composition map is constructed so that the output of \( g \) becomes the input for \( f \). This operation has some noteworthy properties:

• The order of function composition matters; generally, \( f \circ g eq g \circ f \).
• Composing functions can transfer properties such as one-to-oneness from the individual functions to their composition, as proven in the exercise.
• It's a valuable tool in both mathematical theories and real-world applications for simplifying complex operations into manageable pieces.

This concept helps us to streamline processes by performing multiple operations in sequence, effectively reducing potentially complicated scenarios into simpler instances.

Proving with Contradiction

Proof by contradiction is a logical technique used to prove a statement by assuming the opposite and showing that this assumption leads to a contradiction. The process generally involves the following steps:

• Assume the opposite of what you want to prove is true.
• Show that this assumption leads to an impossibility or something false.
• Conclude that the original statement must be true because the opposite can't hold.

In the context of the original exercise about one-to-one functions and their composition:

• We assumed \( (f \circ g)(x_1) = (f \circ g)(x_2) \) and worked to show \( x_1 = x_2 \), using the property that \( f \) and \( g \) are one-to-one.
• This illustration serves as a demonstration of proving that the composite function is one-to-one by showing the assumption leads back to equal inputs, consequently breaking the initial assumption.
• Using contradiction helps solidify understanding by ruling out all possibilities except the one that must logically and consistently hold true.

This proof strategy is broadly used in mathematics to affirm truths by eliminating false assumptions, helping students clear any conceptual ambiguities by engaging with contradictions.