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# handout07 - Math 5020 Handout 7 Cardinal Arithmetic and the...

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Math 5020 Handout 7 Cardinal Arithmetic and the Axiom of Choice Cardinalities without AC 1. Deﬁne the equinumerosity equivalence relation on V by X e Y ⇐⇒ there is a bijection between X and Y . 2. The cardinality of a set X , denoted by | X | e , is the e -equivalence class of X . In other words, | X | e = | Y | e ⇐⇒ there is a bijection between X and Y . 3. Deﬁne | X | e ≤ | Y | e or | Y | e ≥ | X | e if there is a one-to-one function from X into Y . Deﬁne | X | e < | Y | e or | Y | e > | X | e if | X | ≤ | Y | but | X | ̸ = | Y | . The relation is transitive. 4. (Cantor’s Theorem) For every set X , | X | e < |P ( X ) | e . 5. (Cantor–Schr¨ oder–Bernstein) If | A | e ≤ | B | e and | B | e ≤ | A | e , then | A | e = | B | e . 6. The relation < is a partial order (irreﬂexive and transitive). 7. Deﬁne sum , product , and exponentiation of cardinalities by | X | e + | Y | e = | ( X × { 0 } ) ( Y × { 1 } ) | e | X | e · | Y | e = | X × Y | e | X

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## handout07 - Math 5020 Handout 7 Cardinal Arithmetic and the...

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