SEMEPXDOST 2020 - UPLB DOST Scholars' Society Study Notes
Table of Contents
Arithmetic
Sets
Definition of Sets
Notation
Examples
Subsets
Subset Notation
Numerical Sets
Set of All Real Numbers
Set of All Natural Numbers
Set of All Whole Numbers
Set of All Integers
Set of All Rational Numbers
Set of All Irrational Numbers
Set Builder Notation
Interval Notation
Exercises
Basic Applications of Arithmetic
Percentages
Averages
Ratio and Proportion
Definitions
Ratio Formula
Proportion Formula
Examples
Arithmetic
Sets
Definition of Sets
A set is defined as a collection of things where the members of this collection have a common property.
Notation
To represent a set, we list each element (or "member") separated by commas and enclose the entire listing in curly brackets:
Example: {x,y,z} or {x,y,z, …}
Examples
Sets can also consist of various objects. For example:
S = {socks, shoes, watches, shirts, …}
T = {index, middle, ring, pinky}
U = {d, o, s, t, s, e, m, e, p}
Subsets
Definition
A set "X" is said to be a subset of another set "Y" if all elements of set "X" are elements of set "Y".
Subset Notation
The subset relationship is represented mathematically as:
If A is a subset of B, we write ( A
ightarrow B )
Numerical Sets
The following types of numerical sets are covered:
Set of all Integers
Set of all Real Numbers (denoted by ( ext{IR} ))
Set of all Complex Numbers
Set of All Real Numbers
Definition
The set of all real numbers is denoted by ( ext{IR} ). It includes five subsets:
Subsets of Real Numbers
Rational Numbers: Any number that can be expressed as ( \frac{x}{y} ) where ( x ) and ( y ) are integers and ( y \neq 0 ).
Examples include ( 1.375, -3, 0, \frac{1}{2} ).
Irrational Numbers: Numbers that cannot be expressed as a fraction of two integers, such as ( \pi, \sqrt{2}, ) and ( e ).
Set of All Natural Numbers
Definition
The set of natural numbers (also known as counting numbers) is given by:
( {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, …} )
Set of All Whole Numbers
Definition
The set of whole numbers includes all the elements of the natural numbers plus zero:
( {0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, …} )
Set of All Integers
Definition
The set of integers consists of all whole numbers and their negative counterparts:
( {… -3, -2, -1, 0, 1, 2, 3, …} )
Set of All Rational Numbers
Definition
The set of rational numbers comprises numbers that can be expressed as the ratio of two integers:
( { \frac{x}{y} | x, y \in \mathbb{Z}, y \neq 0 } )
Set of All Irrational Numbers
Definition
Irrational numbers are defined as numbers that cannot be expressed as a ratio of two integers.
Set Builder Notation
Definition
Set Builder Notation is a mathematical notation used to describe a set by enumerating its elements or stating the properties that its members must satisfy.
Examples of Set Builder Notation
Sets can be expressed concisely, for instance:
( A = {1, 2, 3, 4, 5} )
( A = {x | x \in \mathbb{N}, \ 1 \leq x \leq 5} )
Interval Notation
Definition
Interval notation is used to express the domain of a function. The symbols used include:
( \cup ) - Union of two sets
( ( ) ) - Open interval
( [ ] ) - Closed interval
Examples of Interval Notation
To represent intervals, examples include:
( {x | x \neq 0} )
( (-2, 3] )
( (-\infty, 0) \cup (0, \infty) )
Exercises
True or False Questions
All integers are whole numbers. (True)
Some rational numbers are integers. (True)
Some whole numbers are irrational numbers. (False)
All integers are rational. (True)
If a number is rational, then it must be a whole number. (False)
No whole numbers are integers. (False)
Translate to Builder Notation
( {2, 4, 6, 8, 10} \rightarrow {x : x \text{ is an even natural number less than 12})
( {2, 3, 5, 7, 11} \rightarrow {x : x \text{ is a prime number less than 12})
( {January, June, July} \rightarrow {x : x \text{ is a month whose name starts with letter J})
Translate to Interval Notation
( {x | 2 < x < 8} \rightarrow (2, 8))
( {x | -5 \leq x < 0} \rightarrow [-5, 0))
( {x | x > 3} \rightarrow (3, \infty))
( {x | x \leq -4} \rightarrow (-\infty, -4] )
Basic Applications of Arithmetic
Percent
Definition
Percentages express how large or small one quantity is relative to another quantity.
Explanation
A percentage is defined as a number or ratio as a fraction of 100:
( 1 ext{%} = \frac{1}{100} )
Calculating Percentages
To calculate a percentage, turn the numbers into a ratio as a fraction of 100. For example, if 8 out of 15 students are boys, the fraction that represents this can be calculated as follows:
Example 1
For the calculation of 25% of 80:
The equation is:
( 25 ext{%} \rightarrow \frac{25}{100} )
Then, multiply:
( \frac{25}{100} \cdot 80 = 20 )
Averages
Definition
The arithmetic mean, or "average", is a measure of the middle or typical value of a dataset.
It is calculated by dividing the sum of a collection of numbers by the number of elements in that collection.
Example 1
To find the average of the numbers in the set {12, 25, 34, 17, 8, 42}:
Total = 12 + 25 + 34 + 17 + 8 + 42 = 138
Count = 6
Average = ( \frac{138}{6} = 23 )
Example 2
A small company with 8 employees has the following hourly wages:
Ratio refers to the quantitative relationship between two amounts, represented typically in the form of a fraction ( a:b ), meaning ( \frac{a}{b} ) where a and b are any two integers.
Proportion
Proportion asserts that two ratios are equal.
Ratio Formula
If we have two quantities or numbers, the formula for calculating the ratio is defined as:
( a:b \rightarrow \frac{a}{b} )
Proportion Formula
For two ratios, if ( a:b ) and ( c:d ), the formula states:
( \frac{a}{b} = \frac{c}{d} )
Here, ( b ) and ( c ) are termed as means, while ( a ) and ( d ) are referred to as extremes.
Example 1
Are ( 4:5 ) and ( 8:10 ) in proportion?
Calculation:
( \frac{4}{5} = 0.8 )
( \frac{8}{10} = 0.8 )
Since both ratios are equal, they are in proportion.
Example 2
Given a ratio of ( 2:3 ) and the sum of the two numbers equals 60:
Let the two numbers be represented by ( 2x ) and ( 3x ).
The total is: ( 2x + 3x = 60 )
Simplifying gives ( 5x = 60 ) leading to ( x = 12 ).