In exercises 1 and 2, determine which of the numbers are `a`. integers, `b`. rational numbers, `c`. irrational numbers, `d`. real numbers, `e`. prime numbers, `f`. composite numbers.

1. `-3     4     1/5     11     3.14     57     0.252252225...`

2. `5.bar17     -4.25     1/4     pi     21     53     0.45454545...`

In exercises 3 to 6, list the elements of the set.

3. `A={x|x` is a composite number less than 11`}`

4. `B={x|x` is an even prime number`}`

5. `C={x|50 < x < 60` and `x` is a prime number`}`

6. `D={x|x` is the smallest odd composite number`}`

In exercises 7 to 12, list the four smallest elements of each infinite set.

7. `{2x|x` is a positive integer`}`

8. `{|x||x` is an integer`}`

9. `{y|y=2x+1`, `x` is a natural number`}`

10. `{y|y=x^2-1`, `x` is an integer`}`

11. `{z|z=|x|`, `x` is an integer`}`

12. `{z|z=|x|-x`, `x` is a negative integer`}`

In exercises 13 to 24, use `A={0,1,2,3,4}`, `B={1,3,5,11}`, `C={1,3,6,10}`, and `D={0,2,4,6,8,10}` to find the indicated intersection or union.
13. `AnnB`	14. `AnnC`	15. `BnnC`
16. `BnnD`	17. `AnnD`	18. `CnnD`
19. `AuuB`	20. `AuuC`	21. `Ann(BuuC)`
22. `Auu(BnnC)`	23. `(BnnC)nnD`	24. `Auu(BuuC)`

In exercises 25 to 38, identify the property of real numbers or the property of equality that is illustrated.
25. `3+(2+5)=(3+2)+5`	26. `6+(2+7)=(6+2)+7`
27. `1*a=a`	28. If `a+b=2`, then `2=a+b`
29. `a(bx)=a(bx)`	30. If `x+2y=7` and `7=y`, then `x+2(7)=7`
31. If `x=2(y+z)` and `2(y+z)=5w`, then `x=5w`	32. `p(q+r)=pq+pr`
33. `m+(-m)=0`	34. `t(1/t)=1`, `t!=0`
35. `7(a+b)=7(b+a)`	36. `8(gh+5)=8(hg+5)`
37. If `x+2y=7` and `w=7`, then `x+2y=w`	38. `5[x+(y+z)]=5x+5(y+z)`

In exercises 39 to 48, use the properties of fractions to perform the indicated operations. State each answer in lowest terms. Assume `a` is a nonzero real number.
39. `(2a)/7-(5a)/7`	40. `(2a)/5+(3a)/7`
41. `(-3a)/5+a/4`	42. `7/8a-13/5a`
43. `(-5)/7*2/3`	44. `7/11*(-22)/21`
45. `(12a)/5-:(-2a)/3`	46. `2/5-:3 2/3`
47. `(2a)/3-(4a)/5`	48. `1/(2a)-3/a`

49. Pool Maintenance. One pipe can fill a pool in 11 hours. A second pipe can fill the same pool in 15 hours. Assume the first pipe fills `1/11` of the pool every hour and the second pipe fills `1/15` of the pool every hour.
a. Find the amount of the pool the two pipes together fill in 3 hours.
b. Find the amount of the pool they fill together in `x` hours.

50. Focal Length of a Mirror. The relationship between the distance of an object `d_0` from a curved mirror, the distance of its image `d_i`, from the mirror, and the focal length `f` of the mirror is given by the mirror equation:

`1/f=1/d_0+1/d_i`
What is the focal length `f` of a mirror² for which `d_0=25` centimeters and `d_i=5` centimeters.

51. State the multiplicative inverse of `7 3/8`.

52. State the multiplicative inverse of `-4 2/5`.

53. Show by an example that the operation of subtraction of real numbers is not a commutative operation.

54. Show by an example that the operation of division of nonzero real numbers is not a commutative operation.

55. Show by an example that the operation of subtraction of real numbers is not an associative operation.

56. Show by an example that the operation of division of real numbers is not an associative operation.

In exercises 57 to 66, classify each statement as true or false.
57. `a/0` is the multiplicative inverse of `0/a`.	58. `(-1/pi)` is the multiplicative inverse of `-pi`.
59. If `p=q+t/2` and `q+t/2=1/2s`, then `1/2s=p`.	60. If `a-b=7`, `7=b-a`.
61. The sum of two composite numbers is a composite number.	62. All integers are natural numbers.
63. Every real number is either a rational number or an irrational number.	64. Every rational number is either even or odd.
65. `1` is the only positive integer that is not prime and not composite.	66. All repeationg decimals are rational numbers.

67. Use a calculator to write each of the following rational numbers as a decimal. If the number is represented by a nonterminating decimal, then use a bar over the repeating portion of the decimal.
a.`8/11`	b.`33/40`	c.`2/7`	d.`5/37`

68. Use a calculator to determine whether `3.14` or `22/7` is a closer approximation to `pi`.

69. Use a calculator to complete the following table.
`x`	`0.1`	`0.01`	`0.001`	`0.0000001`
`(sqrt(x+9)-3)/x`
Now make a guess as to the number the fraction seems to be approaching as `x` assumes values of real numbers closer and closer to zero.

70. Use a calculator to complete the following table.
`x`	`0.1`	`0.01`	`0.001`	`0.0000001`
`(1/2-1/(x+2))/x`
Now make a guess as to the number the fraction seems to be approaching as `x` assumes values of real numbers closer and closer to zero.

71. Which of the properties of real numbers are satisfied by the set of positive integers?

72. Which of the properties of real numbers are satisfied by the set of integers?

73. Which of the properties of real numbers are satisfied by the set of rational numbers?

74. Which of the properties of real numbers are satisfied by the set of irrational numbers?

75. Goldbach's Conjecture. In 1742 Christian Goldbach conjectured that every even number greater than 2 can be written as the sum of two prime numbers. Many mathematicians have tried to prove and disprove this conjecture without succeeding. Show that Goldbach's conjecture is true for the following even numbers
a. 12
b. 30

76. Twin Primes. If the natural numbers `n` and `n+2` are both prime numbers, then they are said to be twin primes. For example, `11` and `13` are twin primes. It is not known whether the set of twin primes is an infinite set or a finite set. List all the twin primes less than 50.