Fundamentals of Mathematical Analysis by Rod Haggarty

By Rod Haggarty

Offering scholars with a transparent and comprehensible creation to the basics of study, this ebook maintains to offer the elemental suggestions of research in as painless a way as attainable. to accomplish this target, the second one variation has made many advancements in exposition.

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5. The difference A - B of two sets A and B is given by A-B=An~B Use the laws of the algebra of sets to establish the follmving: (a) (A - H) u (A - C) =A - (B n C) (b) (A-B)-C=A-(HUC) (c) (A ffi B) - C ~A E0 (B - C) Show by example that equality need not occur in part (c). 6. Prove that for any sets A, 8 and C A x (B 7. n C) = (A x B) n (A x C) The power set of a set A is the set ~(A) consisting of all subsets of A. For example, ~({O, 1}) = {0, {O}, {l}, {O, l} }. P(B) ~ H) (1i(A U B) Show by example that equality need not hold in part (b ).

It is also used to prove that v'2 exists as a real number and hence that the set of rational numbers satisfies all of the axioms for the real numbers with the exception of the completeness axiom. Thus it is the completeness axiom that distinguish:::s between the set 0 of rational numbers and the set ~ of real numbers. 1 Numbers Our first encounter with numbers is with the set N of counting numbers or natural numbers. These are 1, 2, 3, ... and so on. ~th adults leads to an awareness of their significance in phrases such as 'two eyes' and 'four marbles'.

1, where it is shown that the distance usually denoted by \/2 does not correspond to any rational. 1 uses familiar notions such as the decimal expansion of numbers to show that between any two poinL~ on the number line there are infinitely many points that do not correspond to rationals as well a-. infinitely many that do. The existence, at least in geometric terms. of non-rational (or irrational) numbers was a :source of great confusion for early mathematicians, and it wa-. only in the nineteenth century that the entire system of real numbers was successfully developed from the set N of natural numhcrs.

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