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Chapter 10: Problem 9

Define a binary relation \(P\) on \(\mathbf{Z}\) as follows: For all \(m, n \in \mathbf{Z}\), \(m P n \Leftrightarrow m\) and \(n\) have a common prime factor. a. Is \(15 P 25\) ? b. \(22 P 27\) ? c. Is \(0 P 5\) ? d. Is \(8 P 8 ?\)

Short Answer

Expert verified
a. Yes, \(15 P 25\), since they share the common prime factor 5. b. No, \(22 \nrightarrow P 27\), as they don't share any common prime factors. c. Yes, \(0 P 5\), since 0 is divisible by every prime number. d. Yes, \(8 P 8\), since a number will always share a common factor with itself.

Step by step solution

01

Define Prime Factors

Prime factors of a number \(n\) are the set of prime numbers that divide \(n\) exactly. Let's find the prime factors of each number first.
02

Checking for pair (15, 25)

The prime factors of 15 are 3 and 5. The prime factors of 25 are 5. We can see that both numbers have 5 as a common prime factor. Hence \(15 P 25\).
03

Checking for pair (22, 27)

The prime factors of 22 are 2 and 11. The prime factors of 27 are 3. There is no common prime factor. Hence \(22 \nrightarrow P 27\).
04

Checking for pair (0, 5)

Since zero is divisible by every number, it shares a prime factor with every number as well. Hence \(0 P 5\).
05

Checking for pair (8, 8)

A number will always have a common factor with itself. The prime factor of 8 is 2, so \(8 P 8\).

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Most popular questions from this chapter

Let \(P\) be a set of parts shipped to a company from various suppliers, \(S\) is the relation defined on \(P\) as follows: For all \(x, y \in P\), \(x S y \Leftrightarrow x\) has the same part number and is shipped from the same supplier as \(y\).

A number of binary relations are defined on the set \(A=\\{0,1,2,3\\}\). For each relation: a. Draw the directed graph. b. Determine whether the relation is reflexive. c. Determine whether the relation is symmetric. d. Determine whether the relation is transitive. Give a counterexample in each case in which the relation does not satisfy one of the properties. $$R_{7}=\\{(0,3),(2,3)\\}$$

Define a relation \(R\) on the set \(\mathbf{Z}\) of all integers as follows: For all \(m, n \in \mathbf{Z}\), \(m R n \Leftrightarrow m+n\) is even. Is \(R\) a partial order relation? Prove or give a counterexample.

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Determine whether the given binary relation is reflexive, symmetric, transitive, or none of these. Justify your answers. Let \(A\) be the set of people living in the world today. \(A\) binary relation \(R\) is defined on \(A\) as follows: For all \(p, q \in A, p R q \Leftrightarrow p\) lives within 100 miles of \(q\).
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