In mathematics, a set is simply a collection of distinct objects, called elements or members, grouped together because they share some property or characteristic. You can think of it like a "basket" where you collect items that fit a certain rule or idea.
Some other examples:
- All vowels in the English alphabet: {a, e, i, o, u}
- Numbers greater than 5: {5, 6, 7, 8, 9, ...}
- Days in a Week: D = {Monday, Tuesday, Wednesday, Thursday, Friday, Saturday, Sunday}
- Colors in a Traffic Light: T = {red, yellow, green}
Mathematical Definition
Set A is written as: A = {x ∣ property of x}
This means A is the set of all x such that x satisfies a certain property.
Key characteristics
- Well-defined: The contents of the set are specified and identifiable. Ex. {1, 2, 3, 4} is the set of natural numbers less than 5.
- Distinct Elements: A set cannot have duplicate elements. Ex. {1, 2, 2, 3} is the same as {1, 2, 3}.
- Order of Elements: The order of elements does not matter. Ex. {1, 2, 3} is the same as {3, 2, 1}.
Examples of Sets
Finite Sets
- A set of vowels in the English alphabet: A = {a, e, i, o, u}
- A set of natural numbers less than 6: B = {1, 2, 3, 4, 5}
- A set of primary colors: C = {red, blue, yellow}
Infinite Sets
- A set of natural numbers: N = {1, 2, 3, 4, . . .}
- A set of integers: Z = {. . . , -3, -2, -1, 0, 1, 2, 3, . . . }
- A set of real numbers greater than 0: R+ = {x∣ x > 0}
Empty (Null) Set
- A set of months with 32 days: ∅ = {}
- A set of natural numbers less than 1: ∅ = {}
➢Practice: Solved Examples
