course mth.151
Dr. Smith, Please let me know early how I did.
Exercise 2.1Match each set in column I to the appropriate description in column II.
Column I Column II
3. {
, -4,-2,-2,-1} E
5. {2,4,8,16,32} B
6. {
,-5,-3,-1,1,3,5,
} D
9. The set of all counting numbers less than or equal to 6. {1,2,3,4,5,6} Also {X|X are all counting numbers between 1 and 6}
10. The set of all whole numbers greater than 8 and less than 18. {9,10,11,12,13,14,15,16,17,18} Also expressed as : {{X|X are the whole numbers between 8 and 18}
12. Set of all counting numbers between 4 and 14. {X|X are all counting numbers between 4 and 14}. This is also expressed as {4,5,6,7,8,9,10,11,12,13,14}
x is a veritable that can stand for any object. Set-builder notation provides a test for whether a given object is a member of the set.
For example, x = 8, then x is in the set. However it does not satisfy the condition you state, which is 'X are all counting numbers between 4 and 14'. Substituting 8 for x we get
8 are all counting numbers between 4 and 14.
Ignoring the grammatical error (plural verb 'are' applied to singular element 8), the real problem is that 8 is not ' all counting numbers between 4 and 14'.
8 is just one counting number.
A correct application of set-builder notation would be
{X|X is a counting number between 4 and 14}.
Substituting 8 would then yield the correct statement
'8 is a counting number between 4 and 14'.
You make, and should correct similar errors on your subsequent applications of set-builder notation.
15. {-15,-13,-11,
-1} {X|X is all negative odd numbers between -15 and -1} Also expressed as: -15,-13,-11,-9,-7,-5,-3,-1}
18. {90,87,84,
,69} {X|X Are all numbers between 90 and 69 that are multiples of 3}. Also expressed as: {90,87,84,81,78,75,72,69}
20. {X|X is an odd integer between -8 and 7} {-7,-5,-3,-1,0,1,3,5,7} or {-1 is an odd integer between -8 and 7}
Denote each set by listing method .
21. The set of all counting numbers greater than 20. {21,22,23,
.}
24. The set of U.S. Presidents who served after
Lyndon Johnson and before George W. Bush {Nixon,Ford, Carter,Reagan,George H. W. Bush and Bill Clinton}
25.{X|X is a positive multiple of 5} {5,10,15,20,25,
}
27. {X|X is the reciprocral of a natural number} {1,1/2, 1/3, 1/4, 1/5,
.}
Use each set by set builder notation , using X as a variable.
30. The set of all even natural numbers. {X is the set of all even natural numbers greater than 2,
}
Identify each set as finite or infinite.
33. {2,4,6,
32} Finite
35. {1/2, 2/3, Ύ,
} Infinite
36. {-10, -8, -6,
} Infinite
39. {X|X is a natural number greater than 50} Infinite
40. {X|X is a rational number between 0 and 1} Infinite
42. A={-3, -1, 1, 3, 5, 7, 9} 7 A better response would be n(A) = 7; then it would be implicitly clear what the question was.
45.A= {a,b,c,
,z} 26
48. A= the set of current U.S. Senators. 100
50.A={1/2, - ½, 1/3,-1/3,
,1/10, -1/10} 0 It isn't clear what the question here is, but it has to do with cardinality it probably isn't correct.
51. Explain why it is acceptable to write the statement
x is a vowel in the English alphabet in the set for exercise
46 despite the fact that it is a consonant. Answers will vary.
54. X|X is a negative number Well defined
55. {X|X is a good athlete Not well defined
60. 8__Not True____ {3, -2, 5, 7, 8}
63. 0___True {-2, 0, 5, 9}
65. {3} Not True {2,3,4,6}
66. {6} _Not True_ {2+1, 3+1,4+1,5+1,6+1}
69. 3 True {2,5,6,8} This statement is false.
70. 6 is an Element of {-2, 5, 8, 9} This statement is false
72. m is an element of {l,m,n,o,p} This statement is true.
75.{k,c,r,a}={k,c,a,r} This statement is true.
78. {3,7,12,14}= {3,7,12,14,0} Statement is False.
80. 4 is an element of {3} , {4}, {5} } True
81. {X|X is a natural number greater than 10} = {11, 12, 13,
} True
84. 8 is an element of B True
85. 4 is not an element of C False
87. Every element of C is also an element of A True
90. {X|X is a good athlete is not well defined for example and is very ambiguous as to what a good athlete is. However {X|X is a real number} is well defined because a real number is considered to be any number that can be expressed as a decimal. This is more concrete and not as subjective as what is a Good Athlete.
93. Two sets are equivalent if they have the same number of elements but do not have the identical same elements. If they did this would make them equal.
95. The answer to (a) is General Electric, Pfizer, Nortel Networks, and Lucent.
The answer to 95 (b) is General Electric, Motorola, Time Warner, Citigroup, Texas Instruments, EMC corp., and AT&T wireless.
96. (a) Golf and swimming, or canoeing and swimming
(b) Volleyball,Golf, and Running. Running, Swimming, and Volleyball and Golf, canoeing and running.
Exercise 2.2
Column Column
I II
3. {a,b} C
5. U A
6. {a} D
9. {2,5} _is a subset of_ {0,1,5,3,4,2}
10.{a, n, d} is a subset of {r,a,n,d,y}
12. 0 is not a subset of 0.
15. {B, C, D} is both a subset and a proper subset of {B. C, D, F}
18. {S,M,T,W, Th} is a proper subset of {M, W, Th, S} The reverse statement is true, but this is not.
20. The empty set, is a subset of the empty set.
21. {-1, 0, 1, 2, 3} is neither a subset or an improper subset of {0, 1, 2, 3, 4}
24. C is a proper subset of U True or False? True
25. D is a subset of B? False
27. A is a proper subset of B? True
30. The Empty set is a subset of D? False
33. D is not a subset of B? True
35. There are exactly 6 subsets of C? False
36. There are exactly 31 subsets of B? False
39. There is exactly 1 subset of the empty set? True
40. There are exactly 127 proper subsets of U? False
41.The venn Diagram for problem 41 on page 64 correctly represents the relationship among sets A , C, and U? False.
42. The venn diagram for problem 42 page 64 correctly represents the relationship among sets B. C, and U? False
45. {a,b,c,d,e,f} A.64 B. 63
you do not provide enough information for me to understand the reasons for your true and false answers in the above questions; just 'true' or 'false' is never a sufficient answer and is never right; when an answer can be selected from a small list of possibilities, as is the case for true-false questions, a complete statement justifying the answer is required.
48.X|X is an odd whole number less than 4? A. 16 B. 15
50. {2, 5, 7, 9, 10} The complement is {1, 3, 4, 6, 8}
51. Complement of {1. 3, 4, 5, 6, 7, 8, 9, 10} is {2}
54.Complement of set U is: Empty set.
55.Higher cost, Lower cost, Educational, more time to see the sights,Less time to see sights, cannot visit relatives along the way, can visit relatives along the way.
57. Higher cost, more time to see sights, cannot visit relatives along the way.
60. Higher cost, more time to see sights, cannot visit relatives along the way.
63. {A, B, C} , {A, B, D}, {A, B, E}, {A, C, D} {A, C, E} {A, D, E}, {B, C, D}, {B, C, E}, {B, D, E}, {C, D, E}
65. {A}, {B}, {C}, {D}, {E}
66. Empty Set
69. A.15 B. 16
70. A. 15 B. 16
It is clear that you are doing well here. However some of these questions do not contain enough information for me to evaluate your answers.
I have inserted notes on some questions, especially related to set-builder notation. Be sure to review those notes and let me know if you have questions.
This is not the format in which your work is to be submitted. As specified on the Assignments page, the assignment is to submit the SEND files created by the q_a_ and query programs. You will find that the process is much less work than what you have done here.
If you prefer to submit your work in this format, this is fine with me, but I need to see all the information I need in a single document. This means that you will have to provide more information on many of the problems so I can tell what information you are working with, and what is being asked. For example on problems 24-45, I would need to know what the sets are. On problems 55-60, I would need at least a brief description of the situation and the questions being answered.
I note that you have completed most of the Orientation forms, but has not submitted the SEND file created by the q_a_prelim, nor have you submitted the assigned q_a_initial_probs, which also requires the submission of your SEND file. This leads me to believe that you do not understand how to submit your work from the q_a_ and query programs, and perhaps do not understand how to run those programs.
Go back to the instructions for orientation and run the q_a_prelim. If you have run the q_a_ and query programs as assigned, submit the SEND files they have created. If you have not run these programs, run them and submit the SEND files.
If there is an instruction you do not understand, give me all the details and the specifics.