By Kam-Tim Leung, P.H. Cheung

ISBN-10: 9622091814

ISBN-13: 9789622091818

Uncomplicated techniques of quantity thought are mentioned. subject matters contain set idea, mathematical induction, com-binatorics, mathematics, actual numbers, restrict and convergence, and intricate numbers.

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The mapping (: N x N ~ N maps an ordered pair of natural numbers m onto a single natural number by the rule ([(m, n)] = 2 + l (2n + 1). Show that ( is an injective mapping. 7. Let E be the set of all even natural numbers. For each n E N, let ((n) 2n. Prove that the mapping ( : N -+ E is injective. = 34 Fundamental Concepts of Mathematics 8. Prove that the mapping ( : N ((n) ~ =n N defined by - 1 for n ~ 1 and ((0) = 0 is not injective. 9. Prove that the mapping ( : N ~ N defined by ((n) = n + 2 is not surjective.

That if a F b then Ha}, {a, b}} F Hb}, {b, an. Most importantly we can prove the following crucial theorem. 1. Theorem. Let a, b, c and d be objects. Then Ha}, {a,bn = {{c}, {c,d}} ifandonlyifa=candb=d. 9, if a = c and b = d, then Ha}, {a, b}} He}, {c, d}}. Thus the 'if' part of the theorem is trivial. Let us prove the 'only if' part. Assume that Ha}, {a, b}} = He}, {c, dD. By Proof. 9 again, there are the following two possible cases for the elements of these sets: (i) {a} = {c} and {a, b} = {c, d}, or (ii) {a} = {c, d} and {a, b} = {c}.

Example, Prove that 0 + 1 + 2 + ... + n natural numbers n = 2 (n + 1) holds for all n. Proof. Let S be the set of natural numbers n for which the formula o + 1 + 2 + ... + n =2"n (n + 1) holds: S = {n E NI 0 + 1 + 2 ... + n ="2n (n + 1)}. We want to prove that S = N. For this purpose we need only show that conditions 0) and Oi) are satisfied. Condition (i) is trivial. For Oi), let us suppose that k E S. Then k o + 1 + 2 + ... + k + (k + 1) ="2 (k + 1) + (k + 1) = (k ; 1) (k + 2), Therefore (k + 1) E S.

### Fundamental Concepts of Mathematics by Kam-Tim Leung, P.H. Cheung

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