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Chapter 9: Problem 15

Suppose \(A\) is a set with \(m\) elements and \(B\) is a set with \(n\) elements. a. How many binary relations are there from \(A\) to \(B\) ? Explain. b. How many functions are there from \(A\) to \(B\) ? Explain. c. What fraction of the binary relations from \(A\) to \(B\) are functions?

Short Answer

Expert verified
a. There are \(2^{mn}\) binary relations from A to B. b. There are \(n^n\) functions from A to B. c. The fraction of binary relations from A to B that are functions is \(\frac{n^n}{2^{mn}}\).

Step by step solution

01

Understanding binary relations and functions

A binary relation from set A to set B is a subset of the Cartesian product A x B. In simpler terms, it's a collection of ordered pairs where the first elements are from set A and second elements are from set B. A function from set A to set B is a specific type of binary relation, where each element from set A is associated with exactly one element from set B. Now, let's find out the number of binary relations, the number of functions, and the fraction of binary relations that are functions.
02

Finding the number of binary relations from A to B

As a binary relation is a subset of the Cartesian product A x B, we first need to find the size of the Cartesian product. There are m elements in set A and n elements in set B, so there are a total of m*n ordered pairs in the Cartesian product A x B. Now, each binary relation can have any number of ordered pairs from this Cartesian product. So, the total number of binary relations from A to B is equal to the number of all possible subsets of A x B. This can be calculated using the formula 2^(mn), as each pair can either be in the subset or not, resulting in 2^(mn) possible binary relations.
03

Finding the number of functions from A to B

As mentioned earlier, a function from set A to set B is a specific type of binary relation where each element from set A is associated with exactly one element from set B. So, for each element in set A, there are n choices from set B. Since there are m elements in set A, the total number of functions from A to B is equal to the product of all the possible choices, which is n^n.
04

Calculating the fraction of binary relations that are functions

Finally, to find the fraction of binary relations that are functions, we'll divide the number of functions by the number of binary relations: Fraction = (number of functions) / (number of binary relations) Fraction = (n^n) / (2^(mn)) Let's summarize the answers: a. The total number of binary relations from A to B is 2^(mn). b. The total number of functions from A to B is n^n. c. The fraction of binary relations from A to B that are functions is (n^n) / (2^(mn)).

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Most popular questions from this chapter

Exercises 28-35 refer to selection sort, which is another algorithm to arrange the items in an array in ascending order. Algorithm \(9.3 .2\) Selection Sort \([\) The aim of this algorithm is to take an array \(a[1], a[2],\), \(a[3], \ldots, a[n]\) (where \(n \geq 1\) ) and interchange its values if necessary to put them in ascending order. In the first step, the array item with the least value is found, and its value is as- signed to a \([1] .\) In general, in the kth step, \(a[k]\) is compared to each \(a[i]\) for \(i=k+1,2, \ldots, n\). Whenever the value of \(a[k]\) is greater than that of \(a[i]\), the two values are inter- changed. The process continues through the \((n-1)\) st step after which the array items are in ascending order. Input: \(n[a\) positive integer \(], a[1], a[2], a[3], \ldots, a[n][\) an array of data items capable of being ordered \(]\) Algorithm Body: for \(k:=1\) to \(n-1\) for \(i:=k+1\) to \(n\) if \(a[i]

The following is a formal definition for \(O\)-notation, written using quantifiers and variables: \(f(x)\) is \(O(g(x))\) if, and only if, \(\exists\) positive real numbers \(b\) and \(B\) such that \(\forall x>b\), $$ |f(x)| \leq B|g(x)| \text {. } $$ a. Write the formal negation for the definition using the symbols \(\forall\) and \(\exists\). b. Restate the negation less formally without using the symbols \(\forall\) and \(\exists\).

Define binary relations \(R\) and \(S\) from \(\mathbf{R}\) to \(\mathbf{R}\) as follows: $$ \begin{aligned} &R=\\{(x, y) \in \mathbf{R} \times \mathbf{R}|y=| x \mid\\} \text { and } \\ &S=\\{(x, y) \in \mathbf{R} \times \mathbf{R} \mid y=1\\} \end{aligned} $$ Graph \(R, S, R \cup S\), and \(R \cap S\) in the Cartesian plane,

Show that if a function \(f: \mathbf{R} \rightarrow \mathbf{R}\) is increasing. then \(f\) is one-to-one.

. a. Suppose a function \(F: X \rightarrow Y\) is one-to-one but not onto. Is \(F^{-1}\) (the inverse relation for \(F\) ) a function? Explain your answer. b. Suppose a function \(F: X \rightarrow Y\) is onto but not one-to-one. Is \(F^{-1}\) (the inverse relation for \(F\) ) a function? Explain your answer. Draw the directed graphs of the binary relations defined in 23-27 below.
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