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Chapter 9: Problem 13

a. Find all binary relations from \(\\{0,1\\}\) to \(\\{1\\}\). b. Find all functions from \(\\{0,1\\}\) to \(\\{1\\}\). c. What fraction of the binary relations from \(\\{0,1\\}\) to \(\\{1\\}\) are functions?

Short Answer

Expert verified
There are 4 binary relations between the sets {0,1} and {1}, which are: \[\{\}, \{(0,1)\}, \{(1,1)\}, \{(0,1),(1,1)\}\] Out of these relations, only one is a function, which is \[\{(0,1),(1,1)\}\] The fraction of binary relations that are functions is \(\frac{1}{4}\).

Step by step solution

01

Understand binary relations and functions

A binary relation between two sets A and B is a collection of ordered pairs, where the first element of each pair belongs to A and the second element belongs to B. In other words, a binary relation is a subset of the Cartesian product A 脳 B. A function is a special type of binary relation, where each element of A is related to exactly one element of B.
02

Generate the Cartesian product for the given sets

We will first generate the Cartesian product of the sets {0,1} and {1}, which is given by: \[ \{(0,1),(1,1)\} \]
03

Find all binary relations

Since a binary relation is a subset of the Cartesian product, we can find all possible subsets of the Cartesian product from Step 2. There are a total of 2^2 = 4 possible subsets. These binary relations are: \[ \{\}, \{(0,1)\}, \{(1,1)\}, \{(0,1),(1,1)\} \]
04

Find all functions

A function is a binary relation where each element of the domain set {0,1} is related to exactly one element of the codomain set {1}. Looking at the binary relations we found in Step 3, we can identify the following functions: \[ \{(0,1),(1,1)\} \] This is because it is the only binary relation where each element in {0,1} is related to exactly one element in {1}.
05

Calculate the fraction of binary relations that are functions

We found 4 binary relations in Step 3 and 1 function in Step 4. So, the fraction of binary relations from {0,1} to {1} that are functions is: \[ \frac{1}{4} \] In conclusion, there is one function among the four binary relations between the sets {0,1} and {1}, and the fraction of binary relations that are functions is \(\frac{1}{4}\).

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