Finite Cardinality

Set Theory · Axiom Academy

Properties and operations on finite set cardinalities A set A is finite if there exists a bijection between A and 0, 1, ..., n-1 for some natural number n. The number n is unique and is called the cardinality of A. If you place n+1 objects into n boxes, at least one box must contain more than one object. Formal statement: If |A| > |B| and f: A → B, then f is not injective. This simple principle has profound consequences throughout mathematics. For finite sets, we can count elements in a union using the inclusion-exclusion principle: We subtract the intersection to avoid counting shared elements twice. 4. Cardinality of Cartesian Products The Cartesian product creates ordered pairs from two sets. If A has m elements and B has n elements, then A × B has m · n elements. Example: a, b × 1, 2, 3 has 2 · 3 = 6 pairs. The power set P(A) contains all subsets of A, including ∅ and A itself. If |A| = n, then |P(A)| = 2^n. Example: If A = a, b, c , then P(A) has 2³ = 8 subsets. Each element can either be included or excluded from a subset, giving 2 choices per element.

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