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Chapter 2: Problem 3

Let \(A=\\{a, b, c\\}\) and let \(B=\\{0,1\\}\). Find all possible functions from \(A\) to B. Give each function as a set of ordered pairs. (Hint: Every such function corresponds to one of the subsets of \(A\).)

Short Answer

Expert verified
There are 8 functions from \( A \) to \( B \), each represented by a set of ordered pairs.

Step by step solution

01

Understand Function Mapping

A function from set \( A \) to set \( B \) assigns exactly one element from \( B \) (either 0 or 1 in this case) to each element of \( A \) (elements are \( a, b, c \)).
02

Calculate the Number of Functions

Since each element in \( A \) can be mapped to either 0 or 1 in \( B \), and there are three elements in set \( A \), we calculate the total number of functions as \( 2^3 = 8 \), because there are 2 choices for each of the 3 elements.
03

List All Possible Functions

To list all functions, consider all possible combinations of mappings from elements in \( A \) to elements in \( B \):- \( \{(a,0), (b,0), (c,0)\} \)- \( \{(a,0), (b,0), (c,1)\} \)- \( \{(a,0), (b,1), (c,0)\} \)- \( \{(a,0), (b,1), (c,1)\} \)- \( \{(a,1), (b,0), (c,0)\} \)- \( \{(a,1), (b,0), (c,1)\} \)- \( \{(a,1), (b,1), (c,0)\} \)- \( \{(a,1), (b,1), (c,1)\} \)

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Key Concepts

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

Discrete Mathematics
Discrete mathematics is an area of mathematics focused on objects that can assume only distinct, separated values. This branch of mathematics is essential for computer science and is used in various fields.
  • It includes topics such as graph theory, combinatorics, and logic.
  • Discrete mathematics is crucial in algorithm design, programming languages, and cryptography.
In the context of functions from sets, discrete mathematics allows us to deal with mappings between countable and distinct sets, like those in the original exercise. This kind of problem-solving involves finite collections of objects that are non-continuous by nature, showcasing an important aspect of discrete math.
Set Theory
Set theory is a branch of mathematics that studies sets, which are collections of objects. This theory is foundational for understanding and defining functions.
  • A set is often denoted by curly braces, for example, set \( A = \{a, b, c\} \).
  • Sets can include anything: numbers, letters, symbols, etc.
  • Set operations include union, intersection, and complements, among others.
The notion of a function as a subset of ordered pairs is deeply rooted in set theory. For instance, the functions in our example are essentially subsets of the Cartesian product \( A \times B \), where each element from \( A \) is associated uniquely with an element from \( B \). Set theory provides the language and framework to express these relationships effectively.
Function Mapping
Function mapping is a process of associating each element of a set (domain) to exactly one element of another set (codomain). This is often visualized as arrows linking elements from one set to another.
  • The domain is where the function starts; in our example, the set \( A = \{a, b, c\} \).
  • The codomain is the set where the function ends; in the exercise, this is set \( B = \{0, 1\} \).
  • A function is typically described by a set of ordered pairs that maps input to output, such as \( \{(a,0), (b,1), (c,1)\} \).
In function mapping, it's crucial that each element of the domain is paired with one and only one element of the codomain. The exercise helps illuminate this process by showing every possible mapping from \( A \) to \( B \). Function mapping is fundamental in understanding how different sets relate to each other and in extracting meaningful patterns from mathematical structures.

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Most popular questions from this chapter

Consider each of the following relations on the set of people. Is the relation reflexive? Symmetric? Transitive? Is it an equivalence relation? a) \(x\) is related to \(y\) if \(x\) and \(y\) have the same biological parents. b) \(x\) is related to \(y\) if \(x\) and \(y\) have at least one biological parent in common. c) \(x\) is related to \(y\) if \(x\) and \(y\) were born in the same year. d) \(x\) is related to \(y\) if \(x\) is taller than \(y\). e) \(x\) is related to \(y\) if \(x\) and \(y\) have both visited Honolulu.

Prove that composition of functions is an associative operation. That is, prove that for functions \(f: A \rightarrow B, g: B \rightarrow C\), and \(h: C \rightarrow D\), the compositions \((h \circ g) \circ f\) and \(h \circ(g \circ f)\) are equal.

Write each of the following hexadecimal numbers in binary: a) \(0 \times 123\) b) \(0 \times \mathrm{FADE}\) c) \(0 \times 137 \mathrm{~F}\) d) \(0 \times \mathrm{FF} 11\)

Recall that \(\mathbb{N}\) represents the set of natural numbers. That is, \(\mathbb{N}=\\{0,1,2,3, \ldots\\}\). Let \(X=\\{n \in \mathbb{N} \mid n \geq 5\\}\), let \(Y=\\{n \in \mathbb{N} \mid n \leq 10\\}\), and let \(Z=\\{n \in\) \(\mathbb{N} \mid n\) is an even number \(\\}\). Find each of the following sets: a) \(X \cap Y\) b) \(X \cup Y\) c) \(X \backslash Y\) d) \(\mathbb{N} \backslash Z\) e) \(X \cap Z\) f) \(Y \cap Z\) g) \(Y \cup Z\) h) \(Z \backslash \mathbb{N}\)

Give the value of each of the following expressions as a hexadecimal number: a) \(0 \times 73 \mid 0 \times 56 \mathrm{~A}\) b) \(\sim 0 \times 3 \mathrm{FF} 0 \mathrm{~A} 2 \mathrm{FF}\) c) \((0 \times 44 \mid 0 x 95) \& 0 x E 7\) d) \(0 \times 5 \mathrm{C} 35 \mathrm{~A} 7 \& 0 \mathrm{xFF} 00\) e) \(0 \times 5 \mathrm{C} 35 \mathrm{~A} 7 \& \sim 0 \mathrm{xFF} 00\) f) \(\sim(0 \times 1234 \& 0 \times 4321)\)
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