Sets in Discrete Mathematics: Representations, Cardinality, and Set Operations
Learn about sets in discrete mathematics, including different ways to represent sets (roster notation, set-builder notation), the concept of set cardinality, and fundamental set operations. This guide provides a clear introduction to set theory concepts.
Sets in Discrete Mathematics: Representations and Cardinality
What is a Set?
In mathematics, a set is a well-defined collection of distinct objects. These objects are called elements or members of the set. Sets can contain numbers, letters, names, or any other type of object, but the elements within a set must be distinct (no duplicates allowed).
Set Representations
Sets are represented in two main ways:
1. Roster (or Tabular) Form:
Lists all the elements of a set within curly braces { }, separated by commas. For example, the set of even numbers less than 10 is {2, 4, 6, 8}.
2. Set-Builder Form:
Describes the set by specifying the properties its elements must satisfy. This is written in the form {x | P(x)}, where x represents the elements and P(x) is a condition that must be true for x to be an element of the set. For example, the set of all even numbers less than 10 could be written as {x | x is an even number and x < 10}.
Standard Set Notations
| Notation | Meaning |
|---|---|
| x ∈ A | x is an element of set A |
| x ∉ A | x is not an element of set A |
| ∅ | Empty set (a set with no elements) |
| U | Universal set (the set containing all elements under consideration) |
| ℕ | The set of all natural numbers |
| ℤ | The set of all integers |
| ℤ₀ | The set of all non-zero integers |
| ℤ⁺ | The set of all positive integers |
| ℂ, ℂ₀ | The set of all complex numbers, the set of all non-zero complex numbers |
| ℚ, ℚ₀, ℚ⁺ | The set of all rational numbers, the set of all non-zero rational numbers, the set of all positive rational numbers |
| ℝ, ℝ₀, ℝ⁺ | The set of all real numbers, the set of all non-zero real numbers, the set of all positive real numbers |
Cardinality of a Set
The cardinality of a set is the number of elements in the set. For finite sets, this is simply the count of elements. The cardinality of an infinite set is more complex and uses different classifications.
Examples:
- P = {k, l, m, n}: |P| = 4
- A = {0, 2, 4, 6, ...}: A is countably infinite.
Conclusion
Sets are a fundamental concept in discrete mathematics. Understanding how to represent sets and determine their cardinality is essential for working with various mathematical structures.