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Chapter 5: Problem 4

If there are 2187 functions \(f: A \rightarrow B\) and \(|B|=3\), what is \(|A|\) ?

Short Answer

Expert verified
The size of set A is 7.

Step by step solution

01

Write down the formula for number of functions

The number of functions from set A to set B is given by the formula: \(|B|^{|A|}\). In this case, it is given that there are 2187 functions from A to B, and that \(|B| = 3\). So the formula becomes: \(3^{|A|} = 2187\).
02

Solve the equation for \(|A|\)

We need to solve the equation \(3^{|A|} = 2187\) for \(|A|\). Start by writing the number 2187 as a power of 3: 2187 = 3^7. This gives us: \(3^{|A|} = 3^7\), which implies that \(|A| = 7\) due to the rule of exponents. So there are 7 elements in set A.

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Key Concepts

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

Set Theory
In the realm of discrete mathematics, set theory forms the cornerstone of various concepts. It deals with the study of well-defined collections of objects, referred to as sets, and the relationships between them. Sets can be anything from numbers or letters to more complex elements, as long as they can be clearly defined.

When discussing functions like in our example problem, set theory becomes valuable as it provides the formal structure within which functions operate. A function is a specific type of relation that associates each element in a set, known as the domain (typically set A), with exactly one element in another set, called the codomain (typically set B). Understanding these roles of sets is crucial for analyzing functions in discrete math.
Cardinality of Sets
The concept of cardinality refers to the number of elements within a set. Symbolically, we denote the cardinality of a set A as \(|A|\) or simply as 'the number of elements in A'. Cardinality is a measure that allows us to compare different sets regarding their size.

For instance, if set A has two elements and set B has three elements, the cardinality of set A is 2 (so \(|A| = 2\)), and the cardinality of set B is 3 (\(|B| = 3\)). This attribute of sets helps determine the possible number of functions between two sets, as seen in the textbook problem where recognizing the cardinality of set B as 3 led to the identification of the cardinality of set A.
Exponential Functions
Exponential functions play a key role in quantifying the number of functions between two sets. These functions have the form \(b^x\), where \(b\) is the base and \(x\) is the exponent. In the world of sets, they describe scenarios where the choices or possibilities multiply at each step.

For every element in the domain set A, there are \(|B|\) possible elements in set B that it can map to. If the cardinality of set B is 3, as indicated in our problem, then each element of A has three choices in B. The total number of functions is then \(3^{|A|}\), based on the cardinality of A. If we have a cardinality of 7 for A, there are \(3^7\) possible functions. This exponential relationship is fundamental to figuring out the number of functions we can form between two given sets.

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Most popular questions from this chapter

Let \(S\) be a set of seven positive integers the maximum of which is at most 24. Prove that the sums of the elements in all the nonempty subsets of \(S\) cannot be distinct.

The following Pascal program segment implements an algorithm for determining the maximum value in an array \(A[1], A[2], A[3], \ldots, A[n]\) of integers. The array and the value of \(n(\geq 2)\) are supplied earlier in the program; the integer variables \(i\) and Max are declared at the start of the program. (Here the entries in the array need not be distinct) Begin \(\mathrm{Max}:=\mathrm{A}[1]\) For \(1:=2\) to \(n\) do If \(A[1]>M a x\) then Max : = A[1] End; a) If the worst-case complexity function \(f(n)\) for this segment is determined by the number of times the comparison \(A[i]>\mathrm{Max}\) is executed, find the appropriate "big-Oh" form for \(f\). b) What can we say about the best-case and average-case complexities for this implementation?

Let \(f: A \rightarrow A\) be an invertible function. For \(n \in \mathbf{Z}^{+}\) prove that \(\left(f^{n}\right)^{-1}=\left(f^{-1}\right)^{n}\). (This result can be used to define \(f^{-n}\) as either \(\left(f^{n}\right)^{-1}\) or \(\left.\left(f^{-1}\right)^{n} .\right)\)

Let \(f: \mathbf{R} \rightarrow \mathbf{R}\) be defined by \(f(x)=x^{2}\). For each of the following subsets \(B\) of \(\mathbf{R}\), find \(f^{-1}(B)\). a) \(B=\\{0,1\\}\) b) \(B=\\{-1,0,1\\}\) c) \(B=[0,1]\) d) \(B=[0,1)\) e) \(B=[-1,1]\) f) \(B=[0,4]\) g) \(B=[0,1] \cup[4,9]\) h) \(B=(0,1] \cup(4,9)\) Determine three infinite subsets \(B\) of \(\mathbf{R}\) for which \(f^{-1}(B)=\emptyset\).

a) Write a computer program (or develop an algorithm) to locate the first occurrence of the maximum value in an array \(A[1], A[2], A[3], \ldots, A[n]\) of integers. (Here \(n \in \mathbf{Z}^{+}\) and the entries in the array need not be distinct.) b) Determine the worst-case complexity function for the implementation developed in part (a).
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