🔸A set is a collection of well-defined, distinct objects. The objects contained in the set are called elements. A set A is a subset of another set B if its elements are contained in set B.

🔸Set notation is used in mathematics to essentially list numbers, objects, or outcomes. Set notation uses curly brackets { }, which are sometimes referred to as braces. Objects placed within the brackets are called the elements of a set, and do not have to be in any specific order. Sets are named using capital letters, with some sets having predefined names, like N for natural numbers and W for whole numbers.

🔸WAYS OF DESCRIBING A SET

There are two basic methods to describe a set.

A. Verbal Description Method -The set is described in words using a verbal statement.

B. Roster or Listing Method -The elements of the set are listed in a row, separated using commas, and enclosed with braces.

🔸TYPES OF SETS

A. Finite Set - The set has a limited number of elements and can be counted.

B. Infinite Set - The set has an unlimited number of elements, which may or may not be countable.

C. Empty or Null Set - The set has no elements.

D. Universal Set - The set contains all relevant elements for a particular context.

🔸The cardinality of a set is defined as the number of elements in the set. It can be finite or infinite.

🔸Union of Sets

The union of sets A and B, denoted by A ⋃ B, is the set containing all elements that are in A, or in B, or in both A and B.

🔸Intersection of Sets

The intersection of sets A and B, denoted by A ⋂ B, is the set containing all elements that are in both A and B. 

🔸Real numbers are grouped into subsets. These are Counting or Natural Numbers, Whole Numbers, Integers, Rational Numbers, and Irrational Numbers.

🔸Natural/Counting Numbers - The set of numbers used for counting.

🔸Whole numbers - The set of counting numbers including zero.

🔸Integers - The set of natural numbers, their opposites (the negative numbers),and zero.

🔸Rational Numbers - The set of numbers that can be expressed as a fraction a/b, where a and b are integers and 𝑏 ≠ 0.

🔸Irrational Numbers - The set of numbers that cannot be written as a fraction a/b, where 𝑎 and 𝑏 are integers and 𝑏 ≠ 0 or simply, the numbers that are not rational.

🔸Perfect square - The number of square units that can form a square.

🔸Perfect cube - The number of cube units that can form a cube is called a.

🔸The square root of the area of the square (perfect square) is the length of the side of the square.

🔸The cube root of the volume of a cube (perfect cube) is the length of each side of the cube.

🔸When a number n is multiplied by itself, such as when we compute the area of a square, we write n², and read it “n squared”. The result is called the square of n. That is, if n² = 𝑚, then m is a square of n and m is a perfect square.

🔸When a number is cubed, we write n³ and read it “n cubed”. The result is called the cube of n. That is, if n³ = 𝑚, then m is a cube of n and m is a perfect cube.

🔸Irrational numbers are numbers that cannot be expressed as a fraction of two integers (where the numerator and denominator are both whole numbers, and the denominator is not zero). In other words, they cannot be written in the form p/q, where p and q are integers.

🔸If the radicand of a square root is not a perfect square, then it is considered an irrational number. Likewise, if the radicand of a cube root is not a perfect cube, then it is an irrational number. These numbers cannot be written as a fraction because the decimal does not end (or non-terminating) and does not repeat a pattern (or non-repeating).

🔸In plotting an irrational number involving square root or cube root on a number line, estimate first the square root or cube root of the given irrational number and to which two consecutive integers it lies in between.

🔸Measurement is a fundamental aspect of human civilization, and its development has played a crucial role in shaping our understanding of the world. From ancient civilizations to the modern scientific era, the evolution of measurement has been a fascinating journey.

🔸Non-standard units of measurement, sometimes referred to as informal or customary units, have a rich history in human civilization. These units are unique because they lack the standardized and universally recognized measurements found in systems like the Metric and English systems. Instead, non-standard units often reflect the ingenuity and adaptability of people throughout time. They are based on everyday objects, body parts, or localized traditions, making them both fascinating and challenging when it comes to precise measurement. Non-standard units of measure can vary from one person to another or from one region to another. These unconventional units continue to influence our daily lives and how we understand the world.

🔸The English System of Measurement is also known as the Imperial System, a historical and culturally significant system of measurement that has been used primarily in the United States, Liberia, and Myanmar, and to a lesser extent, in the United Kingdom. This system is characterized by its unique units for measuring length, weight, and capacity, which can be quite different from the standardized metric system used in many other parts of the world. Understanding this system is essential for practical purposes, especially when navigating everyday tasks in regions where it continues to be the prevailing measurement system.

🔸SI (International System of Units) is also known as the Metric System, it is a globally accepted system of measurement that is decimal-based or based on powers of 10. Understanding the SI units and the principles behind them is essential for effective communication and cooperation in today's interconnected world.

🔸Volume is the amount of space taken up or occupied by an object or that is enclosed within a container. The volume measures how much space an object occupies or the capacity of a container.