Read e-book online Elementary Analysis. Volume 1 PDF

By K. S. Snell, J. B. Morgan, W. J. Langford and E. A. Maxwell (Auth.)

ISBN-10: 0080107826

ISBN-13: 9780080107820

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Additional resources for Elementary Analysis. Volume 1

Sample text

2 "x P is a given point in the plane. PM, PN are the perpendiculars from P to OX, O Y. The coordinates of P are the directed lengths 34 ELEMENTARY ANALYSIS VOLUME 1 OM = x and ON = y. We write P is the point (x, y), or, more briefly, P is (x, y). Here (x, j ) is an ordered pair of real numbers. The set of all ordered pairs (x, y) is in one-one correspondence with the set of points in the plane. If X = {x} and Y = {>>}, the set of ordered pairs (x, y) is called the cartesian product of X and Y, and is written X X Y.

5* + 7 == 2 (mod 10). 7. 3JC - 5 = 7 (mod 15). 8. 4x + 5 = 3 (mod 10). 50 ELEMENTARY ANALYSIS VOLUME 1 9. 8JC - 1 = 4 (mod 23). 10. 2x - 1 = 2 (mod 4). 11. By considering the results of nos. 2-10, state the conditions for the congruence ax + b = c (mod m) to have (i) a unique solution, (ii) no solution. 12. If n is an integer, find possible values of (i)« 2 (mod5), (ii) n2 (mod 10), (iii) /i3 (mod 10). 13. Prove that 38 - 1 is divisible by 80. 14. 4 8 + 106 = 4 (mod 9). 15. Show that the product of four consecutive integers is divisible by 24, and deduce that n(n + 23)(/i + 25)(n + 74) = 0 (mod 24).

Identity Numbers 0 and 1 exist such that α + 0==0 + α = α, α χ 1 = 1 χ α = α. 48 ELEMENTARY ANALYSIS VOLUME 1 5. Inverse Corresponding to a there exist numbers denoted by (—a) and AT1 such that: a + (-a) = (-a) + a = 0, a x (a-1) = (AT1) X a = l (mod «), provided that n is prime and a f£ 0 (mod w). 6. Distributive 0(6 + c) = ab + ac, (a + b) c = ac + be. The rule of cancellation for addition presents no difficulty, since, for any integer x, a + x = b + xo(a + x) — (b + x) =0oa — b =0 and so a+x =b+ x o a ==b.

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Elementary Analysis. Volume 1 by K. S. Snell, J. B. Morgan, W. J. Langford and E. A. Maxwell (Auth.)


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