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Chapter 5: Problem 30

Give a recursive definition of each of these sets of ordered pairs of positive integers. \([\text {Hint} : \text { Plot the points in the set in }\) the plane and look for lines containing points in the set. \(]\) a) \(S=\left\\{(a, b) | a \in \mathbf{Z}^{+}, b \in \mathbf{Z}^{+}, \text { and } a+b \text { is odd }\right\\}\) b) \(S=\left\\{(a, b) | a \in \mathbf{Z}^{+}, b \in \mathbf{Z}^{+}, \text { and } a | b\right\\}\) c) \(S=\left\\{(a, b) | a \in \mathbf{Z}^{+}, b \in \mathbf{Z}^{+}, \text { and } 3 | a+b\right\\}\)

Short Answer

Expert verified
a) Start with (1,2) and apply (a+2k, b) or (a, b+2k). b) Start with (1,1) and apply (a, b+ak) or (a+k, b). c) Start with (1,2) and apply (a+3k, b) or (a, b+3k).

Step by step solution

01

Understand the problem

We need to provide a recursive definition for each set of ordered pairs, implying that each set can be generated by repeatedly applying a rule or a formula. First, identify the pattern in the properties given for each set.
02

Consider set a) S={(a,b) | a鈭圸+, b鈭圸+, and a+b is odd}

For the set S, the sum of any pair (a, b) must be odd. This means one integer must be odd and the other must be even (e.g., (1, 2), (3, 4)).
03

Define initial pair for set a)

Choose an initial pair such as (1,2). This pair fulfills the condition where the sum is odd (1 + 2 = 3).
04

Recursive rule for set a)

Include a recursive rule that ensures any pair generated has an odd sum: If (a, b) is in the set and a is odd, generate new pairs (a+2k, b) or (a, b+2k) where k is a positive integer. This maintains the condition of the sum being odd.
05

Consider set b) S={(a,b)| a鈭圸+, b鈭圸+, and a|b}

For the set S, a must be a divisor of b. This can start with initial pairs where b is a multiple of a.
06

Define initial pair for set b)

Choose an initial pair such as (1, 1). Here, 1 divides 1 and follows the given condition.
07

Recursive rule for set b)

If (a, b) is in the set, generate new pairs of the form (a, b+ak) or (a+k, b) where k is a positive integer. This ensures that a will always divide b.
08

Consider set c) S={(a,b) | a鈭圸+, b鈭圸+, and 3|a+b}

For this set S, the sum of the pair (a, b) must be divisible by 3.
09

Define initial pair for set c)

Choose an initial pair such as (1, 2). Their sum is 3, which is divisible by 3.
10

Recursive rule for set c)

If (a, b) is in the set, generate new pairs (a+3k, b) or (a, b+3k) where k is a positive integer. This keeps a+b divisible by 3.

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Key Concepts

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

Positive Integers
Positive integers are numbers greater than zero and do not include fractions, decimals, or negative numbers. They are represented as \(\textbf{Z^+}\), which denotes the set of all positive whole numbers, such as 1, 2, 3, and so on. These numbers are fundamental in mathematics because they are used in counting and ordering.
  • Example: The numbers 1, 2, and 3 are positive integers.
  • Use: Positive integers are often used for indexing or labeling items in a list.
Positive integers are crucial when dealing with sets of ordered pairs, as they form the basis for defining recursive rules in these sets.
Ordered Pairs
An ordered pair is a pair of elements \((a, b)\), where the order of the elements is significant. In the context of positive integers, an ordered pair consists of two positive integers, \(a\) and \(b\).For example:
  • (1, 2) is an ordered pair where 1 is the first element and 2 is the second.
  • (2, 1) is different from (1, 2) because the order of elements matters.
Ordered pairs are visualized as points on a plane, with the first element representing the x-coordinate and the second the y-coordinate. In set definitions, the properties of these pairs, such as sums or divisibility, are often explored.
Recursive Rules
Recursive rules define how elements of a set can be built from one or more initial elements using a specific rule repeatedly. For example, to define a set of ordered pairs, you might start with an initial pair and then apply the same rule again and again to generate new pairs.Steps for using recursive rules:
  • Identify an initial element or pair that satisfies the conditions of the set.
  • Define the rule that generates new elements or pairs from existing ones.
  • Apply the rule iteratively to form the entire set.
For instance, with the set \(\textbf{S = \( \{(a, b) | a, b \in \mathbf{Z^+}, \text{and} \ a + b \text{is odd} \} \)}\), you start with an initial pair (like (1, 2)) and apply the rule to keep generating pairs that meet the set's condition (for example, \(a + 2k\) where k = 1,2,3,...).
Divisibility
Divisibility in mathematics means that one integer can be divided by another without leaving a remainder. For example, a positive integer \(b\) is divisible by another positive integer \(a\) if there exists an integer \(k\) such that \(b = a \cdot k\).
  • Example: 10 is divisible by 2 (since 10 = 2 \cdot 5).
  • Non-example: 7 is not divisible by 3 (since 7/3 does not produce a whole number).
In the context of ordered pairs, we can define sets where one element is divisible by the other. For instance, in the set \(\textbf{S = \( \{(a, b) | a, b \in \mathbf{Z^+}, \text{and} \ a | b \} \)}\), every pair (a, b) must satisfy the condition that \(a\) divides \(b\).
Odd and Even Sums
The sum of two positive integers can be categorized as either odd or even. Understanding this helps define sets based on the sum's properties. Here's how it works:
  • An odd number is not divisible by 2.
  • An even number is divisible by 2.
Combining Odd and Even Numbers:
  • \textbf{Odd + Odd = Even}: For example, 1 + 1 = 2.
  • \textbf{Even + Even = Even}: For example, 2 + 2 = 4.
  • \textbf{Odd + Even = Odd}: For example, 1 + 2 = 3.
In the set \(\textbf{S = \( \{(a, b) | a, b \in \mathbf{Z^+}, \text{and} \ a + b \text{is odd} \} \)}\), you need one odd and one even number for the sum to be odd. Thus, if the sum is odd, it ensures that you'll continue generating pairs with an odd sum by alternately adding even increments.

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Most popular questions from this chapter

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Suppose there are \(n\) people in a group, each aware of a scandal no one else in the group knows about. These people communicate by telephone; when two people in the group talk, they share information about all scandals each knows about. For example, on the first call, two people share information, so by the end of the call, each of these people knows about two scandals. The gossip problem asks for \(G(n),\) the minimum number of telephone calls that are needed for all \(n\) people to learn about all the scandals. Exercises \(69-71\) deal with the gossip problem. Use mathematical induction to prove that \(G(n) \leq 2 n-4\) for \(n \geq 4 .\) [Hint: In the inductive step, have a new person call a particular person at the start and at the end. \(]\)

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