Problem 1 - Applied Discrete Structures

Question

Let \([G ; *]\) be a group and \(a\) be an element of \(G .\) Define \(f: G \rightarrow G\) by \(f(x)=a * x\) (a) Prove that \(f\) is a bijection. (b) On the basis of part a, describe a set of bijections on the set of integers.

Step-by-Step Solution

Answer

The function \( f(x) = a * x \) is a bijection because it's injective and surjective by group properties. On integers, similar mappings like \( f(n) = a + n \) are bijections.

Step 1: Define Bijection

To prove that a function is a bijection, we need to show that it is both injective (one-to-one) and surjective (onto). We'll start by defining the function: \( f: G \rightarrow G \) where \( f(x) = a * x \). We are tasked with demonstrating that this mapping is a bijection.

Step 2: Prove Injectiveness

For \( f \) to be injective, if \( f(x_1) = f(x_2) \), then \( x_1 = x_2 \). Assume \( f(x_1) = f(x_2) \), which implies \( a * x_1 = a * x_2 \). Since \([G; *]\) is a group, every element has an inverse. Thus, multiply both sides on the left by \( a^{-1} \) (inverse of \( a \)): \( a^{-1} * (a * x_1) = a^{-1} * (a * x_2) \), simplifying this gives \( x_1 = x_2 \). Therefore, \( f \) is injective.

Step 3: Prove Surjectiveness

For \( f \) to be surjective, for every \( y \in G \), there exists an \( x \in G \) such that \( f(x) = y \). Take any \( y \in G \). Do we have \( x \) such that \( a * x = y \)? Choose \( x = a^{-1} * y \). Then, \( f(x) = a * (a^{-1} * y) = (a * a^{-1}) * y = e * y = y \), where \( e \) is the identity element in the group. Thus, \( f \) is surjective.

Step 4: Conclude Bijection Proof

Since \( f \) is both injective and surjective, \( f \) is a bijection. This means \( f \) establishes a one-to-one correspondence between the elements of \( G \) and \( G \).

Step 5: Number Bijections in Integers

Given the function as defined above, similarly, bijections on the set of integers \( \mathbb{Z} \) can be formed by taking any integer \( a \) and defining \( f(n) = a + n \) for \( n \in \mathbb{Z} \). With addition of inverses in \( \mathbb{Z} \), these functions are bijections, mapping each integer to another uniquely.

Key Concepts

Bijection

A bijection is a special type of function. It connects every element of one set to another set with no leftovers and no duplicates.
- **Injective Function**: Each element in the first set maps to a unique element in the second set. No two different elements have the same image.- **Surjective Function**: Every element in the second set is paired with at least one element from the first set.If a function is both injective and surjective, it's a bijection. In group theory, establishing that a function like \( f(x) = a * x \) is a bijection involves showing these properties.

Injective Function

An injective function, also known as a one-to-one function, is where each element in the domain maps to a distinct and unique element in the codomain. If two elements of the domain are mapped to the same element in the codomain, those elements must be the same.
For example, consider the group \([G; *]\). To prove the function \( f(x) = a * x \) is injective, assume \( f(x_1) = f(x_2) \), then it must be true that \( x_1 = x_2 \). Here, using the property of the inverse element in groups, multiplying both sides by \( a^{-1} \), simplifies the equation to show \( x_1 = x_2 \). This injectiveness ensures no two elements map to the same outcome.

Surjective Function

A surjective function, or onto function, is one where every element in the codomain is the image of at least one element from the domain. This means the function covers the whole target set.
In the group setting, for the function \( f(x) = a * x \) to be surjective, for any element \( y \) in the group \( G \), there should exist an element \( x \) in \( G \) such that \( f(x) = y \).
Choosing \( x = a^{-1} * y \), ensures \( f(x) = y \), where \( a^{-1} \) is the inverse element and \( e \) is the identity element. This demonstrates that every possible output has a pre-image, confirming it's surjective.

Inverse Element

The concept of an inverse element is crucial in group theory. In a group \([G; *]\), every element \( a \) has an inverse \( a^{-1} \), such that their operation results in the identity element \( e \).
For example, in a group operation, performing \( a * a^{-1} = e \) clears the operation to the identity. This property is key to proving injectivity and surjectivity because it helps revert operations back.
In the discussed function, the existence of aggregate inverse operations allows us to prove both injectiveness and surjectiveness, hence establishing function \( f(x) = a * x \) as a bijection. Inverses thus allow the creation of unique correspondences between elements of a group.