By Saharon Shelah

Is the continuum speculation nonetheless open? If we interpret it as discovering the legislation of cardinal mathematics (or exponentiation, due to the fact that addition and multiplication have been classically solved), the speculation will be solved by way of the independence result of Godel, Cohen, and Easton, with a few remoted confident effects (like Gavin-Hajnal). such a lot mathematicians anticipate that merely extra independence effects stay to be proved. In Cardinal mathematics, even though, Saharon Shelah bargains an alternate view. by way of redefining the speculation, he will get new effects for the traditional cardinal mathematics, reveals new functions, extends older tools utilizing basic filters, and proves the lifestyles of Jonsson algebra. Researchers in set conception and comparable parts of mathematical common sense probably want to learn this provocative new method of an immense subject.

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2 # 14 7 3 61. 4b a , 9 27 65. 3x 9 , 4 16y a Z 0 y Z 0 58. 2# 9 3 10 62. 3a b , 7 21 66. 14m # 4 2 7 b Z 0 59. 2 10 , 7 3 63. 3x 6x , 10 15 x Z 0 64. 4 1 1 , 7 5 20 67. 3y 6x , 7 28 y Z 0 68. 2 1# 5 7 3 6 In Exercises 69–72, evaluate the algebraic expression for the speciﬁed values. 69. 71. ■ -c 2d for c ϭ Ϫ4, d ϭ 3 m1 # m2 r 2 for m1 ϭ 3, m2 ϭ 4, r ϭ 10 70. 2l ϩ 2w 72. x - m s for for l ϭ 5, w ϭ 10 x ϭ 100, m ϭ 70, s ϭ 15 A P P L I C AT I O N S On December 16, 2007, the United States debt was estimated at $9,176,366,494,947, and at that time the estimated population was 303,818,361 citizens.

2 Ϫ 3[(4x Ϫ 5) Ϫ 3x Ϫ 7] - 14 5 - (- 2) -3 (5)(- 1) 32. - 35. Ϫ(6x Ϫ 4y) Ϫ (3x ϩ 5y) 39. - 4(5) - 5 -5 36. 12 (-3)(-4) -4x 6 - (- 2) 40. Ϫ6(2x ϩ 3y) Ϫ [3x Ϫ (2 Ϫ 5y)] In Exercises 41–56, write as a single fraction and simplify. 41. 1 5 + 3 4 42. 1 1 2 5 43. 5 1 6 3 44. 7 1 3 6 45. 3 5 + 2 12 46. 1 5 + 3 9 47. 1 2 9 27 48. (-4) 3 5 7 3 6 49. x 2x + 5 15 50. y y 3 6 51. x 2x 3 7 52. y y 10 15 53. 4y (- 3y) 15 4 54. 6x 7x 12 20 55. 3 7 + 40 24 56. -3 -7 - a b 10 12 60. 4 7 , 5 10 In Exercises 57–68, perform the indicated operation and simplify, if possible.

2)3 # 52 = (-2)(-2)(-2) # 5 # 5 = Evaluate the expression Ϫ43 # 23. So far, we have discussed only exponents that are natural numbers (positive integers). When the exponent is a negative integer, we use the following property. N E GATIVE - I NTE G E R ■ Answer: Ϫ512 Study Tip A negative exponent implies a reciprocal. E X P O N E NT P R O P E RT Y Let a be any nonzero real number and n be a natural number (positive integer); then a-n = 1 an a Z 0 In other words, a base raised to a negative-integer exponent is equivalent to the reciprocal of the base raised to the opposite (positive) integer exponent.