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Chapter 2: Problem 16

Consider these functions from the set of students in a discrete mathematics class. Under what conditions is the function one-to-one if it assigns to a student his or her a) mobile phone number. b) student identification number. c) final grade in the class. d) home town.

Short Answer

Expert verified
The functions are one-to-one if each student has a unique mobile phone number or student ID. They are not one-to-one if the function assigns final grades or home towns.

Step by step solution

01

Understanding one-to-one functions

A function is one-to-one (injective) if each element of the domain is mapped to a unique element of the codomain. In other words, no two different elements in the domain map to the same element in the codomain.
02

Part a: Mobile Phone Number

For a function assigning a student's mobile phone number to be one-to-one, each student must have a unique mobile phone number. No two students should share the same mobile phone number.
03

Part b: Student Identification Number

For a function assigning a student's identification number to be one-to-one, each student must have a unique student ID. No two students should share the same student ID number.
04

Part c: Final Grade

For a function assigning a student's final grade to be one-to-one, each student must receive a unique final grade. Since students often receive the same grades (e.g., multiple students achieving an 'A'), this function is typically not one-to-one.
05

Part d: Home Town

For a function assigning a student's home town to be one-to-one, each student must come from a different home town. Since students from the same class can come from the same home town, this function is usually not one-to-one.

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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, also known as a one-to-one function, ensures that each element in the domain maps to a distinct and unique element in the codomain.
In simpler terms, no two different items in the start set (domain) should end up being the same item in the end set (codomain).
Mathematically, this means if we have a function \( f: A \rightarrow B \), then for any \(x_1, x_2 \in A\): \[ f(x_1) = f(x_2) \implies x_1 = x_2 \]
This property prevents any overlap in the mapping, ensuring a clear and distinct relationship between the domain and codomain elements.
Unique Mapping
Unique mapping is a key characteristic of one-to-one functions. Each input (from the domain) should have its own distinct output (in the codomain).
For a function to be injective, no two inputs should map to the same output.
Let's break down the exercises from the textbook:
  • a) Mobile Phone Number: The function is injective if each student has a unique mobile number. No two students should share the same number.
  • b) Student Identification Number: The function is one-to-one if every student has a distinct ID number. Shared IDs would make it non-injective.
  • c) Final Grade: This function is generally not one-to-one, as multiple students can receive the same grade (e.g., multiple 'A' grades).
  • d) Home Town: Similarly, this function is usually not injective, as multiple students can originate from the same hometown.
Discrete Mathematics
Discrete mathematics is a core field of study within mathematics that deals with discrete elements.
It includes topics such as set theory, graph theory, and combinatorics.
Understanding one-to-one functions is crucial in discrete mathematics, as they are fundamental in the study of functions and relations.
Key areas where injective functions play a critical role include:
  • Algorithm design: Ensuring optimal, non-redundant mappings in data structures.
  • Database management: Unique identifier assignments to avoid data conflicts.
  • Graph theory: Establishing unique vertex/edge mappings for effective graph representation and analysis.
This concept helps students understand how elements can be distinctly mapped, aiding in better data organization and problem-solving strategies.

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Most popular questions from this chapter

Let \(A\) and \(B\) be sets. Show that a) \((A \cap B) \subseteq A\) b) \(A \subseteq(A \cup B)\) c) \(A-B \subseteq A\) d) \(A \cap(B-A)=\emptyset\) e) \(A \cup(B-A)=A \cup B\)

Give an example of a function from \(\mathbf{N}\) to \(\mathbf{N}\) that is a) one-to-one but not onto. b) onto but not one-to-one. c) both onto and one-to-one (but different from the identity function). d) neither one-to-one nor onto.

List the first 10 terms of each of these sequences. a) the sequence that begins with 2 and in which each successive term is 3 more than the preceding term b) the sequence that lists each positive integer three times, in increasing order c) the sequence that lists the odd positive integers in in- creasing order, listing each odd integer twice d) the sequence whose nth term is \(n !-2^{n}\) e) the sequence that begins with 3, where each succeeding term is twice the preceding term f ) the sequence whose first term is 2, second term is 4, and each succeeding term is the sum of the two preceding terms g) the sequence whose nth term is the number of bits in the binary expansion of the number n (defined in Section 4.2) h) the sequence where the nth term is the number of letters in the English word for the index n

Consider these functions from the set of teachers in a school. Under what conditions is the function one-to-one if it assigns to a teacher his or her a) office. b) assigned bus to chaperone in a group of buses taking students on a field trip. c) salary. d) social security number.

Show that if \(A, B,\) and \(C\) are sets, then \(\overline{A \cap B \cap C}=\overline{A} \cup$$\overline{B} \cup \overline{C}\) a) by showing each side is a subset of the other side. b) using a membership table.
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