Cardinality Definition

Set Theory · Axiom Academy

Understanding when two sets have the same size 1. Equal Cardinality via Bijection Two sets A and B have the same cardinality if there exists a bijection (one-to-one correspondence) between them. The relation "has the same cardinality as" is an equivalence relation on the class of all sets. Reflexive: Every set has the same cardinality as itself (identity function). Symmetric: If |A| = |B|, then |B| = |A| (inverse of bijection). Transitive: If |A| = |B| and |B| = |C|, then |A| = |C| (composition of bijections). For finite sets, cardinality matches our intuitive notion of counting. A finite set with n elements has cardinality n. If A = a, b, c , then |A| = 3 because there's a bijection with 0, 1, 2 . 4. Cardinal Numbers as Equivalence Classes A cardinal number is an equivalence class of sets under the "same cardinality" relation. All sets with 3 elements belong to the equivalence class we call "3". Let's verify that different-looking sets can have the same cardinality: Example 1: | a, b, c | = | 1, 2, 3 | = 3 Example 2: | red, green, blue | = | △, ○, □ | = 3 All of these sets are in bijection with each other, so they share the same cardinal number.

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