define and identify types of sets
define and identify types of real numbers and their properties
Sets:
A set is a collection of distinct objects, called elements, which are represented within curly brackets {}. Sets can consist of various types of elements, such as numbers, letters, or even other sets.
Types of sets:
1. Finite set: A set that contains a specific number of elements. For example, the set {1, 2, 3} is a finite set since it has three elements.
2. Infinite set: A set that contains an infinite number of elements. For example, the set of all natural numbers {1, 2, 3, ...} is an infinite set.
3. Empty set (or null set): A set with no elements. It is denoted by the symbol Ø or {}.
4. Singleton set: A set that contains only one element. For example, the set {5} is a singleton set.
5. Subset: A set that contains elements that are all part of another set. For example, if A = {1, 2, 3} and B = {1, 2}, then B is a subset of A because all elements of B are part of A.
Real Numbers:
Real numbers are a set of numbers that include rational numbers (numbers that can be expressed as a fraction) and irrational numbers (numbers that cannot be expressed as a fraction). Real numbers can be represented on the number line.
Types of real numbers:
1. Natural numbers (N): The set of positive whole numbers: 1, 2, 3, ...
2. Whole numbers (W): The set of non-negative whole numbers, including zero: 0, 1, 2, 3, ...
3. Integers (Z): The set of all whole numbers, including negative numbers: ..., -3, -2, -1, 0, 1, 2, 3, ...
4. Rational numbers (Q): Numbers that can be expressed as a fraction of two integers: 1/2, -3/4, 6, 7/2, etc.
5. Irrational numbers (I): Numbers that cannot be expressed as a fraction and have non-repeating decimal representations, such as √2, π, or e.
Properties of real numbers:
Real numbers possess the following properties:
1. Closure property: The sum or product of any two real numbers is always a real number.
2. Associative property: For addition and multiplication of real numbers, changing the grouping of the numbers does not affect the result. (a + b) + c = a + (b + c) and (a • b) • c = a • (b • c).
3. Commutative property: For addition and multiplication of real numbers, the order of the numbers does not affect the result. a + b = b + a and a • b = b • a.
4. Identity property: There exists an identity element such that adding or multiplying it with any real number does not change the number. For addition, the identity element is 0, and for multiplication, the identity element is 1.
5. Inverse property: For every real number, there exists its additive and multiplicative inverses. The additive inverse of a is -a, and the multiplicative inverse of a, if it exists, is 1/a.