Introduction to Sets in Discrete Mathematics

📚 What is Discrete Mathematics?

Discrete Mathematics is the study of mathematical structures that are countable or otherwise distinct and separable — such as combinations, graphs, and logical statements. These structures can be finite or infinite.

🔠 What are Sets?

A set is an unordered collection of distinct objects, called elements. If an object a belongs to a set A, we write a ∈ A. If not, a ∉ A.

Examples:

Vowels = { a, e, i, o, u }
Students = { Fred, Tom, Mary, Jane }

For infinite or large sets, use ellipsis:

A = {1, 2, 3, 4, 5, ...}

📏 Cardinality of a Set

Cardinality refers to the number of elements in a set. It’s denoted as |A|. Finite sets have natural number cardinality; infinite sets are just referred to as infinite.

🚫 The Empty Set

An empty set contains no elements. It’s written as {} or ∅. Note: { ∅ } is a set with one element — the empty set.

📂 Subsets

If all elements of set A are in set B, then A ⊆ B. If B has extra elements, A is a proper subset: A ⊂ B. If both sets contain exactly the same elements: A = B.