100 Thus 1277 875 X <
Elements Of Electromagnetics
7th Edition
ISBN:9780190698614
Author:Sadiku, Matthew N. O.
Publisher:Sadiku, Matthew N. O.
ChapterMA: Math Assessment
Section: Chapter Questions
Problem 1.1MA
Related questions
Question
3, 5, 9,19, 21
Transcribed Image Text:11:27 MO
←
100
Thus, 1277 875twelve-
=
=
96 Numeration Systems and Whole Number Operations
Example 5
X <
Assessment 3-1A
1. Write the following numerals in words.
a. 56,283,914
b. 5,365,295,234
2. Indicate which digit in 234,518 has place value
a. hundreds
b. ten thousands.
3. For each of the following base-ten numerals, give the place
value of the underlined digit.
a. 827,367
b. 8,421,000
4. Rewrite each of the following as a base-ten numeral.
a. 3-10 +4-10³ +5
b. 2.10 + 1
5. If , A, and O each represent different non-zero base-ten
digits, order the numerals from least to greatest.
a. D; ADO;DA
b. ΑΠΟ: ΠΠΔΔ
6. How many digits are there in the standard form of the
numeral representing 625-10¹2
7. A certain three-digit natural number has the following
properties: The hundreds digit is greater than 7; the tens
digit is an odd number; and the sum of the three digits is 10.
What could the number be?
8. For each of the following pairs of numerals tell which
numeral represents the greater number.
a. MCDXXIV and MCDXXIV
b. 4632 and 46,032
c. <Y and < TV
d. 9990nil and I
e.
and
9. For each of the following numerals write both the succeeding
and the preceding numerals (one more and one less).
a. MCMXLIX
c. £99
b.くくく▼
d.
14. Write the base-four numeral for the representation
shown.
466330
15. Write the first 15 counting numerals for each of the
following bases.
a. Two
b. Four
Hiffonded for th
|||
Rob used base twelve to write the following:
g36twelve = 1050
What is the value of g? Check your answer.
Solution We could write the following equations:
g-12 +3.12 + 6-1 = 1050
144g + 36 + 6 = 1050
144g +42
1050
144g = 1008
8=7
Check 736twelve = 7.12²+3•12 + 6-1 = 1050
LTE
LTE1 426%
661 of 1286
O
>
10. A cornerstone representing when a building was built reads
MCMXXII. When was this building built?
11. Write each of the following numerals in Roman symbols.
a. 121
b. 42
d. 2014
c. 91
12. Complete the following table, which compares symbols for
numbers in different numeration systems:
Hindu-
Arabic Babylonian Egyptian Roman Mayan
72
b.
< VY
C.
99nnill
13. In the counting frame, the value of each dot is represented
by the number in the box below the dot. For example, the
following figure represents the number 154.
64
8
1
What numbers are represented in the following frames?
a.
25
5
1
b.
.
8
4
2
1
Section 3-1 Numeration Systems
97
write the corresponding number in the given base in the
following.
a. Ten flats in base ten
b. Twenty flats in base twelve
26. Convert each of the following base-ten numerals to a
numeral in the indicated bases.
a. 456 in base five.
b. 1782 in base twelve
c. 32 in base two
27. Write each of the following numerals in base ten.
a. 432
b. 101101
![11:30 M
64
26%
2
4, or 2²
3
8, or 2¹
Table 11 suggests that for four elements, there might be 2¹, or 16, subsets. Is this correct
If S = {a,b,c,d), then all the subsets of R = {a,b,c} are also subsets of S. Eight new subscr
are also formed by including the element d in each of the eight subsets of R. The eight new
subsets are {d}, {a,d}, {b,d}, {c,d}, {a,b,d}, {a, c, d]}, {b,c,d), and (a, b, c, d). Thus,
there are twice as many subsets of set S (with four elements) as there are of set R (with three
elements). Consequently, there are 2-8, or 2, subsets of a set with four elements. Because in-
cluding one more element in a finite set doubles the number of possible subsets of the new set,
a set with five elements will have 2-2, or 25, subsets, and so on. In each case, the number of
elements and the power of 2 used to obtain the number of subsets are equal. Therefore, if there
are n elements in a set, 2" subsets can be formed. If we apply this formula to the empty set; that is,
when n = 0, we have 20 = 1. The pattern is meaningful because the empty set has only one
subset-itself.
NOW TRY THIS 6
a. How many proper subsets does a set with five elements have?
b. Write a formula to show how many proper subsets a set with z elements has. Explain your
reasoning.
10. Suppose C is a subset of D and D is a subset of C.
a. If n(C) = 5, find (D).
b. What other relationship exists between sets C and D?
11. If A
a, b, c, d, c),
a. how many subsets does / have?
b. how many proper subsets does A4 have?
c. how many subsets does A have that include the elements a
and e?
12. If a set has 127 proper subsets, how many elements are in the
set?
13. Identify all the possible proper subset relationships that
occur among the following sets.
A = (3n|n EN), B = {6n|n=N},
C
(12n
EN}.
14. Indicate which symbol, E or, makes each of the following
statements true.
a. 0
(xx 2 and nEN)
b. 1024
c. 3002
(xx= 3n-1 and n EN}
d. x (xx 2" and " EN}
15. Indicate which symbol, Cor , makes each part of
problem 14 true.
16. Answer each of the following. If your answer is no, tell why.
a. If A = B, can we always conclude that AC B?
b. If AC B, can we always conclude that AC B?
c. If ACB, can we always conclude that A CB?
d. If ACB, can we always conclude that A= B?
17. Use the definition of less than to show each of the following.
a. 3 < 100
b. 0 < 3
18. On a certain senate committee there are seven senators: Abel,
Brooke, Cox, Dean, Eggers, Funk, and Gage. Three of these
members are to be appointed to a subcommittee. How many
possible subcommittees are there?
19. Name two infinite sets that are equivalent but not equal.
20. Write an argument to show that the set of even natural
numbers and the set of odd natural numbers should have the
same cardinal number.
21. Draw a Venn diagram showing the relationship between the
set of natural numbers and the set of whole numbers.
22. Draw a Venn diagram depicting the Adamsville Beta Club,
the officers of the Beta Club, and a member Sanna who is not
an officer of the Beta Club.
23. If the set of officers of the Adamsville Beta Club is equivalent
to the set of members of the Adamsville Beta Club, what can
you infer?
Section 2-2 Describing Sets 65
Introduction to Logic and Sets
Assessment 2-2A
1. Write the following sets using the listing (roster) method or
using set-builder notation.
a. The set of letters in the word assessment
b. The set of natural numbers greater than 20
2. Rewrite the following using mathematical symbols.
a. P is equal to the set containing p, q, r, and s.
b. The set consisting of the elements 1 and 2 is a proper
subset of (1, 2, 3).
c. The set consisting of the elements 0 and 1 is not a subset
of (1,2,3).
3. Which of the following pairs of sets can be placed in one-to-
one correspondence?
a. (1,2,3,4,5) and (m, n, o, p, q)
b. (a, b, c, d, e, f,...,m} and
(1, 2, 3, 4, 5, 6,
13)
c. [xlr is a letter in the word mathematics) and
(1, 2, 3, 4,
11)
4. How many one-to-one correspondences are there between
two sets with
a. 6 elements each?
b. # elements each? Explain your reasoning.
5. How many one-to-one correspondences are there between
the sets (x,y,z, u, v) and (1, 2, 3, 4, 5} if in each
correspondence
a. x must correspond to 5?
b. x must correspond to 5 and y to 1?
c. x, y, and a must correspond to odd numbers?
6. Which of the following represent equal sets?
B = {x,y,z, w)
A= {a,b,c,d)
C = (c,d, a, b}
E=0
D = {x|1 ≤ x ≤ 4 and x EN)
F = {Ø)
G = {0}
H = { }
I= {x|x = 2n + 1 and n EW}
L = (x|x = 2n-1 and EN
7. Find the cardinal number of each of the following sets.
Assume the pattern of elements continues in each part in the
order given.
a. (201, 202, 203,..., 1100)
b. (1, 3, 5,..., 101)
e. (1, 2, 4, 8, 16,..., 1024}
d. (x|x = k¹, k = 1,2,3,..., 100)
8. If U is the set of all college students and A is the set of all
college students with a straight-A average, describe A.
9. Suppose B is a proper subset of C.
=
a. If n(C) 8, what is the maximum number of elements
in B?
b. If n (B) = 8, what is the maximum number of elements
in C?
=
Vo)) LTE
87 of 1044
O
||](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F50012057-e812-4cae-989d-d0b9d24bc771%2F0f308417-3fed-4db1-b2f2-6dda6cbfbb26%2F94egvrv_processed.jpeg&w=3840&q=75)
Transcribed Image Text:11:30 M
64
26%
2
4, or 2²
3
8, or 2¹
Table 11 suggests that for four elements, there might be 2¹, or 16, subsets. Is this correct
If S = {a,b,c,d), then all the subsets of R = {a,b,c} are also subsets of S. Eight new subscr
are also formed by including the element d in each of the eight subsets of R. The eight new
subsets are {d}, {a,d}, {b,d}, {c,d}, {a,b,d}, {a, c, d]}, {b,c,d), and (a, b, c, d). Thus,
there are twice as many subsets of set S (with four elements) as there are of set R (with three
elements). Consequently, there are 2-8, or 2, subsets of a set with four elements. Because in-
cluding one more element in a finite set doubles the number of possible subsets of the new set,
a set with five elements will have 2-2, or 25, subsets, and so on. In each case, the number of
elements and the power of 2 used to obtain the number of subsets are equal. Therefore, if there
are n elements in a set, 2" subsets can be formed. If we apply this formula to the empty set; that is,
when n = 0, we have 20 = 1. The pattern is meaningful because the empty set has only one
subset-itself.
NOW TRY THIS 6
a. How many proper subsets does a set with five elements have?
b. Write a formula to show how many proper subsets a set with z elements has. Explain your
reasoning.
10. Suppose C is a subset of D and D is a subset of C.
a. If n(C) = 5, find (D).
b. What other relationship exists between sets C and D?
11. If A
a, b, c, d, c),
a. how many subsets does / have?
b. how many proper subsets does A4 have?
c. how many subsets does A have that include the elements a
and e?
12. If a set has 127 proper subsets, how many elements are in the
set?
13. Identify all the possible proper subset relationships that
occur among the following sets.
A = (3n|n EN), B = {6n|n=N},
C
(12n
EN}.
14. Indicate which symbol, E or, makes each of the following
statements true.
a. 0
(xx 2 and nEN)
b. 1024
c. 3002
(xx= 3n-1 and n EN}
d. x (xx 2" and " EN}
15. Indicate which symbol, Cor , makes each part of
problem 14 true.
16. Answer each of the following. If your answer is no, tell why.
a. If A = B, can we always conclude that AC B?
b. If AC B, can we always conclude that AC B?
c. If ACB, can we always conclude that A CB?
d. If ACB, can we always conclude that A= B?
17. Use the definition of less than to show each of the following.
a. 3 < 100
b. 0 < 3
18. On a certain senate committee there are seven senators: Abel,
Brooke, Cox, Dean, Eggers, Funk, and Gage. Three of these
members are to be appointed to a subcommittee. How many
possible subcommittees are there?
19. Name two infinite sets that are equivalent but not equal.
20. Write an argument to show that the set of even natural
numbers and the set of odd natural numbers should have the
same cardinal number.
21. Draw a Venn diagram showing the relationship between the
set of natural numbers and the set of whole numbers.
22. Draw a Venn diagram depicting the Adamsville Beta Club,
the officers of the Beta Club, and a member Sanna who is not
an officer of the Beta Club.
23. If the set of officers of the Adamsville Beta Club is equivalent
to the set of members of the Adamsville Beta Club, what can
you infer?
Section 2-2 Describing Sets 65
Introduction to Logic and Sets
Assessment 2-2A
1. Write the following sets using the listing (roster) method or
using set-builder notation.
a. The set of letters in the word assessment
b. The set of natural numbers greater than 20
2. Rewrite the following using mathematical symbols.
a. P is equal to the set containing p, q, r, and s.
b. The set consisting of the elements 1 and 2 is a proper
subset of (1, 2, 3).
c. The set consisting of the elements 0 and 1 is not a subset
of (1,2,3).
3. Which of the following pairs of sets can be placed in one-to-
one correspondence?
a. (1,2,3,4,5) and (m, n, o, p, q)
b. (a, b, c, d, e, f,...,m} and
(1, 2, 3, 4, 5, 6,
13)
c. [xlr is a letter in the word mathematics) and
(1, 2, 3, 4,
11)
4. How many one-to-one correspondences are there between
two sets with
a. 6 elements each?
b. # elements each? Explain your reasoning.
5. How many one-to-one correspondences are there between
the sets (x,y,z, u, v) and (1, 2, 3, 4, 5} if in each
correspondence
a. x must correspond to 5?
b. x must correspond to 5 and y to 1?
c. x, y, and a must correspond to odd numbers?
6. Which of the following represent equal sets?
B = {x,y,z, w)
A= {a,b,c,d)
C = (c,d, a, b}
E=0
D = {x|1 ≤ x ≤ 4 and x EN)
F = {Ø)
G = {0}
H = { }
I= {x|x = 2n + 1 and n EW}
L = (x|x = 2n-1 and EN
7. Find the cardinal number of each of the following sets.
Assume the pattern of elements continues in each part in the
order given.
a. (201, 202, 203,..., 1100)
b. (1, 3, 5,..., 101)
e. (1, 2, 4, 8, 16,..., 1024}
d. (x|x = k¹, k = 1,2,3,..., 100)
8. If U is the set of all college students and A is the set of all
college students with a straight-A average, describe A.
9. Suppose B is a proper subset of C.
=
a. If n(C) 8, what is the maximum number of elements
in B?
b. If n (B) = 8, what is the maximum number of elements
in C?
=
Vo)) LTE
87 of 1044
O
||
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