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Chapter 3: Problem 43

The identity function \(1_{X}\) is bijective.

Short Answer

Expert verified
The identity function \(1_{X}\) on a set X is bijective. To show this, we proved that \(1_{X}\) is injective by showing that if \(1_{X}(x_1) = 1_{X}(x_2)\), then \(x_1 = x_2\). We also proved that \(1_{X}\) is surjective by showing that for every element \(y \in X\), there exists some \(x \in X\) such that \(1_{X}(x) = y\). Since \(1_{X}\) is both injective and surjective, it is bijective.

Step by step solution

01

Prove that \(1_{X}\) is injective (one-to-one)

To show that \(1_{X}\) is injective, we need to prove that for any \(x_1, x_2 \in X\), if \(1_{X}(x_1) = 1_{X}(x_2)\) then \(x_1 = x_2\). Suppose that \(1_{X}(x_1) = 1_{X}(x_2)\) for some \(x_1, x_2 \in X\). By the definition of the identity function, we know that \(1_{X}(x_1) = x_1\) and \(1_{X}(x_2) = x_2\). So, we have \(x_1 = x_2\). Thus, the identity function \(1_{X}\) is injective.
02

Prove that \(1_{X}\) is surjective (onto)

To show that \(1_{X}\) is surjective, we need to prove that for every element \(y \in X\), there exists some \(x \in X\) such that \(1_{X}(x) = y\). Let \(y\) be an arbitrary element in the set X. We need to find an element \(x \in X\) such that \(1_{X}(x) = y\). Since \(1_{X}\) is the identity function on X, we know that \(1_{X}(y) = y\). Therefore, the element x we were looking for is just \(x = y\). So, the identity function \(1_{X}\) is surjective.
03

Conclusion

We have shown that the identity function \(1_{X}\) on the set X is both injective and surjective. Therefore, the identity function \(1_{X}\) is bijective, as required.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Injective Function
An injective function, often referred to as a one-to-one function, is a fundamental concept in discrete mathematics. The main characteristic of an injective function is that each element of the function's domain maps to a unique element in the codomain. In other words, no two different elements in the domain result in the same output in the codomain. This is an important property when determining the nature of mathematical relationships.

To visualize this, imagine assigning one chocolate bar to each person in a room. If an injective property is maintained, every person would have a distinct chocolate bar, different from everyone else's. If any two people ended up with the same type of bar, the relationship between people and chocolate bars would no longer be injective.
Surjective Function
A surjective function, also known as an onto function, describes another fundamental relationship in mathematics, ensuring that every element in the function's codomain is mapped at least once by an element from the domain. The implication here is that the function covers the entire range without leaving any element behind.

Imagine a game of musical chairs where everyone has a seat once the music stops. A surjective property would mean there are no empty chairs; each chair (element in the codomain) has at least one person (element in the domain) corresponding to it. This contrasts with an injective function, where the focus is on individuals not sharing a chair, whereas in a surjective function, it's about having all chairs taken.
Identity Function
The identity function is a simple yet important concept in discrete mathematics. It is defined on a set X and represented as \(1_{X}\), where every element of the set maps to itself. This means that the identity function takes an input and returns an output that is exactly the same. It's like giving every person in a group a name tag with their own name; everyone is labeled as themselves.

The identity function always produces a straight line y = x when plotted on a graph, which signifies the most basic form of equality between the domain and codomain. It has the unique property of being both injective and surjective. Every element is mapped to itself (injective), and every possible output is achieved (surjective), making it bijective. This is pivotal for understanding more complex functions and transformations in discrete mathematics.
Discrete Mathematics
Discrete mathematics is the branch of mathematics dealing with objects that take on distinct, separated values. It includes studying topics like logic, sets, graph theory, and algorithms and is essential for the fields of computer science and information theory.

One of the reasons discrete mathematics is so valuable is that it equips students and professionals with tools to solve problems related to discrete structures, which are the types of structures that power computer algorithms and networks. Think of it like learning the alphabet of a language before writing stories; discrete mathematics provides the foundational language for many technological applications, from coding a website to securing data communication across networks.

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Most popular questions from this chapter

Let \(f: X \rightarrow Y\) and \(A, B \subseteq X^{\dagger} .\) Prove each. If \(\mathrm{B} \subseteq \mathrm{A} \subseteq \mathrm{X},\) then \(f(\mathrm{A})-f(\mathrm{B}) \subseteq f(\mathrm{A}-\mathrm{B}).\)

(Easter Sunday) The date for Easter Sunday in any year \(y\) can be computed as follows. Let \(a=y \bmod 19, b=y \bmod 4, c=y \bmod 7, d=(19 a+24)\) \(\bmod 30, e=(2 b+4 c+6 d+5) \bmod 7,\) and \(r=(22+d+e) .\) If \(r \leq 31,\) then Easter Sunday is March \(r ;\) otherwise, it is April \([r(\bmod 31)] .\) Compute the date for Easter Sunday in each year. $$2076$$

Determine if the functions are bijective. If they are not bijective, explain why. \(f: \mathbb{R} \times \mathbb{R} \rightarrow \mathbb{R} \times \mathbb{R}\) defined by \(f(x, y)=(x,-y).\)

Let \(x, y \in \mathbb{R} .\) Let \(\max \\{x, y\\}\) denote the maximum of \(x\) and \(y,\) and \(\min \\{x, y\\}\) denote the minimum of \(x\) and \(y .\) Prove each. $$\max \\{x, y\\}-\min \\{x, y\\}=|x-y|$$

Evaluate each sum. $$\sum_{k=-1}^{3}(3 k)^{2}$$
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