Cardinal.pdf - REAL ANALYSIS CARDINAL NUMBERS We use S for...

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REAL ANALYSIS CARDINAL NUMBERS We use S for the cardinal number of a set S . I S T (or T S ) is to mean “ a 1-1 correspondence between S and a subset of T ” (not necessarily a proper subset). II S = T is to mean “ a 1-1 correspondence between S and T . [ < is to mean but not =] We have that: (i) The definitions are reasonable when applied to finite sets. (ii) (a) is transitive, i.e. X Y Y Z X Z (b) = is transitive X = Y Y = Z X = Z = is symmetric S = T T = S = is reflexive S = S (iii) (Bernstein’s Lemma) S T T S S = T (iv) For any two sets either S T or T S . A set S is said to be enumerable (denumerable, countable) ⇔ ∃ a 1-1 corre- spondence between S and the set of all natural numbers. χ 0 is called the cardinal number of the set of all natural numbers. 1. If S χ 0 either S is finite or S = χ 0 2. If S = χ 0 S can be put in 1-1 correspondence with proper subset of itself. 3. Any infinite subset contains an enumerable subset. 1
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4. If S = χ 0 and T is infinite then S T = T . 5. A set is infinite it can be put into 1-1 correspondence with a proper subset of itself. Proof of A Suppose U and V are such that U = V = χ 0 . Then U = u 1 u 2 u 3 . . . V = v 1 v 2 v 3 . . . U V = u 1 v 2 u 2 v 2 . . . = W = w 1 w 2 w 3 w 4 . . . therefore U V = χ 0 .
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