# The Axiom of Choice and Its Implications in Mathematics

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• 1 Elementary Set Theory
1 Elementary Set Theory Notation: fg enclose a set. f1; 2; 3g = f3; 2; 2; 1; 3g because a set is not deﬁned by order or multiplicity. f0; 2; 4;:::g = fxjx is an even natural numberg because two ways of writing a set are equivalent. ; is the empty set. x 2 A denotes x is an element of A. N = f0; 1; 2;:::g are the natural numbers. Z = f:::; −2; −1; 0; 1; 2;:::g are the integers. m Q = f n jm; n 2 Z and n 6= 0g are the rational numbers. R are the real numbers. Axiom 1.1. Axiom of Extensionality Let A; B be sets. If (8x)x 2 A iﬀ x 2 B then A = B. Deﬁnition 1.1 (Subset). Let A; B be sets. Then A is a subset of B, written A ⊆ B iﬀ (8x) if x 2 A then x 2 B. Theorem 1.1. If A ⊆ B and B ⊆ A then A = B. Proof. Let x be arbitrary. Because A ⊆ B if x 2 A then x 2 B Because B ⊆ A if x 2 B then x 2 A Hence, x 2 A iﬀ x 2 B, thus A = B. Deﬁnition 1.2 (Union). Let A; B be sets. The Union A [ B of A and B is deﬁned by x 2 A [ B if x 2 A or x 2 B. Theorem 1.2. A [ (B [ C) = (A [ B) [ C Proof. Let x be arbitrary. x 2 A [ (B [ C) iﬀ x 2 A or x 2 B [ C iﬀ x 2 A or (x 2 B or x 2 C) iﬀ x 2 A or x 2 B or x 2 C iﬀ (x 2 A or x 2 B) or x 2 C iﬀ x 2 A [ B or x 2 C iﬀ x 2 (A [ B) [ C Deﬁnition 1.3 (Intersection).
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• The Axiom of Choice and Its Implications
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• 17 Axiom of Choice
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