Problem 1 - Applied Discrete Structures

Question

List four elements of each of the following sets: (a) \(\\{k \in \mathbb{P} \mid k-1\) is a multiple of 7\(\\}\) (b) \(\\{x \mid x\) is a fruit and its skin is normally eaten \(\\}\) (c) \(\left\\{x \in \mathbb{Q} \mid \frac{1}{x} \in \mathbb{Z}\right\\}\) (d) \(\\{2 n \mid n \in \mathbb{Z}, n<0\\}\) (e) \(\\{s \mid s=1+2+\cdots+n\) for some \(n \in \mathbb{P}\\}\)

Step-by-Step Solution

Answer

(a) 8, 15, 22, 29. (b) Apple, pear, grape, cherry. (c) 1, -1, 1/2, -1/2. (d) -2, -4, -6, -8. (e) 1, 3, 6, 10.

Step 1: Understanding Set (a)

For set (a), we need elements from the set of natural numbers (\(\mathbb{P}\)) where \(k-1\) is a multiple of 7. This implies that \(k = 8, 15, 22, 29, \ldots\). We list the first four elements.

Step 2: List Elements for Set (a)

The first four elements of the set are 8, 15, 22, and 29.

Step 3: Understanding Set (b)

For set (b), we need to list fruits that are typically consumed with their skins. Examples include apples, pears, grapes, and cherries.

Step 4: List Elements for Set (b)

Four elements of the set are apple, pear, grape, and cherry.

Step 5: Understanding Set (c)

For set (c), elements are rational numbers such that when inverted, they yield an integer, i.e., \(\frac{1}{x} = n\) for some integer \(n\). Therefore, \(x = \frac{1}{n}\). We'll use n = 1, -1, 2, -2.

Step 6: List Elements for Set (c)

The elements are 1, -1, \(\frac{1}{2}\), and \(-\frac{1}{2}\).

Step 7: Understanding Set (d)

For set (d), we look for even numbers derived from negative integers \(n < 0\). Examples include -2, -4, -6, and -8.

Step 8: List Elements for Set (d)

Four elements of the set are -2, -4, -6, and -8.

Step 9: Understanding Set (e)

Set (e) includes numbers that can be expressed as the sum of the first \(n\) natural numbers, following the formula \(s = \frac{n(n+1)}{2}\). Examples include 1, 3, 6, and 10.

Step 10: List Elements for Set (e)

The first four elements of the set are 1, 3, 6, and 10.

Key Concepts

Natural Numbers

Natural numbers are the building blocks of mathematics. They include all positive whole numbers starting from 1. So, 1, 2, 3, and so on are natural numbers. We denote the set of natural numbers with the symbol \( \mathbb{N} \).
Natural numbers are generally used for counting and ordering
(e.g., 1st, 2nd, 3rd).

• Examples: 1, 2, 3, 10, 100
• Uses: Counting objects (e.g., five apples), ordinal numbers (e.g., first in line)

It’s important to note these numbers do not include zero, fractions, or negative numbers. This makes them distinct from other number sets like integers or rational numbers.

Rational Numbers

Rational numbers are numbers that can be expressed as the ratio of two integers, where the denominator is not zero. These include fractions, integers, and terminating or repeating decimals.
We denote the set of rational numbers as \( \mathbb{Q} \).
The general form of a rational number is \( \frac{p}{q} \), where \( p \) and \( q \) are integers, and \( q eq 0 \).

• Examples: \( \frac{1}{2} \), \( -3 \), \( 4.5 \) (because it can be written as \( \frac{9}{2} \))
• Key Feature: A rational number becomes an integer when the fraction has a denominator of 1 (e.g., \( \frac{6}{1} = 6 \))

Rational numbers are all about division. Anything that can be divided and expressed as a fraction falls into this category.

Integers

Integers encompass whole numbers that can be positive, negative, or zero. This is what sets them apart from natural numbers, which are only positive, and rational numbers, which include fractions. When we talk about integers, we use the symbol \( \mathbb{Z} \).
The set of integers is infinite and looks somewhat like this:

• Examples: -3, -2, -1, 0, 1, 2, 3...

One key idea in integers is symmetry. For every positive integer, there’s a negative counterpart.
Also, zero holds a central place as it is neither positive nor negative.

Summation Formula

The summation formula is a helpful mathematical tool to calculate the sum of the first \( n \) natural numbers. This can be expressed as: \[S = \frac{n(n + 1)}{2}\]The formula efficiently calculates large sums without manually adding numbers one by one.
For example, if you want to find the sum of the first five natural numbers, plug \( n = 5 \) into the formula: \[S = \frac{5(5 + 1)}{2} = 15\]This quick calculation eliminates counting errors in larger sums.

• Use Case: Generates triangular numbers like 1, 3, 6, 10...
• Another Feature: This is a foundation for arithmetic series and sequences.

The concept of summation is all about efficiency and accuracy in mathematical operations.