Types of Sets

The empty set (also called the null set) is the unique set that contains no elements at all. It is written as ∅ or as {}. For example, the set of real numbers x satisfying x² = −1 is empty, because no real number squared gives a negative result. Key fact: the empty set is a subset of every set. That is, for any set A, ∅ ⊆ A. The cardinality (size) of the empty set is |∅| = 0. Be careful: {∅} is NOT the empty set — it is a set containing one element, which happens to be the empty set.

A finite set is a set whose elements can be counted using the natural numbers 1, 2, 3, … and the counting process terminates. If a set A has exactly n elements, we write |A| = n, and n is called the cardinality of A. For example, A = {2, 4, 6, 8, 10} is a finite set with |A| = 5. The set of days in a week is finite with cardinality 7. Every finite set has exactly 2ⁿ subsets — a result you will use frequently when studying the power set.

An infinite set is one whose elements cannot be placed in a one-to-one correspondence with any finite set {1, 2, …, n}. The set of natural numbers ℕ = {1, 2, 3, …}, the integers ℤ, the rational numbers ℚ, and the real numbers ℝ are all infinite. Georg Cantor's revolutionary work showed that not all infinite sets have the same 'size': ℕ, ℤ, and ℚ are countably infinite (denoted ℵ₀), while ℝ is uncountably infinite — a strictly larger infinity. At high school level, it is enough to recognise that a set described with '…' extending forever is infinite.

Two sets A and B are equal, written A = B, if and only if they contain exactly the same elements. The order in which elements are listed and repetitions do not matter. Formally: A = B ⟺ (∀x)(x ∈ A ↔ x ∈ B). For example, {1, 2, 3} = {3, 1, 2} = {1, 1, 2, 3} because every element of each is in the other. To prove A = B, mathematicians usually show A ⊆ B and B ⊆ A simultaneously. Do not confuse equal sets with equivalent sets — equivalent sets merely have the same cardinality but may contain different elements.

The universal set U (sometimes written Ω) is the set of all objects under consideration in a given context. It acts as the 'universe of discourse.' For example, when studying whole-number factors of 12, a natural choice is U = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12}. When studying real-valued functions, U = ℝ. The complement of any set A is defined relative to U: A' = {x ∈ U : x ∉ A}. The universal set is context-dependent — it must always be stated or clearly implied in a problem.

• A, B — Sets (—): Any two sets being compared for equality.
• \iff — If and only if (—): Biconditional: the statement is true in both directions.
• \subseteq — Subset or equal (—): A ⊆ B means every element of A is also an element of B.
• \land — Logical AND (—): Both conditions must hold simultaneously.

Problem. Let U = {x ∈ ℤ : 1 ≤ x ≤ 10}, A = {2, 4, 6, 8, 10}, and B = {x ∈ U : x is even}. (a) State the type of each set. (b) Determine whether A = B. (c) Find A', the complement of A in U.

Answer. A = B (the sets are equal); A' = {1, 3, 5, 7, 9}.

• Subsets and Power Sets — Every type of set is analysed through the subset relation — understanding subsets is the natural next step after classifying sets.
• Set Operations: Union, Intersection and Complement — Operations such as complement are defined relative to the universal set, making set types essential background knowledge.
• Venn Diagrams — Venn diagrams visually represent the universal set and its subsets, making them a key tool for reasoning about set types.
• Cardinality and Countability — The distinction between finite, countably infinite, and uncountably infinite sets is formalised through the concept of cardinality.