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mathematics
linear algebra
Questions and Answers of
Linear Algebra
For each of the following, fill in the blank with the word converse, inverse, or contrapositive so that the result is a true statement. (a) The converse of the inverse of p → q is the
When she is about to leave a restaurant counter, Mrs. Albanese sees that she has one penny, one nickel, one dime, one quarter, and one half-dollar. In how many ways can she leave some (at least one)
Let A = {1, 2, 3, 4, 5, 7, 8, 10, 11, 14, 17, 18}. a) How many subsets of A contain six elements? b) How many six-element subsets of A contain four even integers and two odd integers? c) How many
Let S = {1, 2, 3, . . . , 29, 30}. How many subsets A of S satisfy(a) |A| = 5?(b) |A| = 5 and the smallest element in A is 5?(c) |A| = 5 and the smallest element in A is less than 5?
(a) How many subsets of {1, 2, 3, . . . , 11} contain at least one even integer? (b) How many subsets of {1, 2, 3,. . . . . . . . . , 12} contain at least one even integer? (c) Generalize the results
Write the next three rows for the Pascal triangle shown in Fig. 3.4
Complete the proof of Theorem 3.1. Theorem 3.1. Let A, B, C ⊂ μ. (a) If A ⊂ B and B ⊂ C, then A ⊂ C. (b) If A ⊂ B and B ⊂ C, then A ⊂ C. (c) If A ⊂ 5 and B ⊂ C, then A ⊂ C. (d)
In part (i) of Fig. 3.5 we have the first six rows of Pascal's triangle, where a hexagon centered at 4 appears in the last three rows. If we consider the six numbers (around 4) at the vertices of
(a) Among the strictly increasing sequences of integers that start with 1 and end with 7 are: (i) 1, 7 (ii) 1, 3, 4, 7 (iii) 1, 2, 4, 5, 6, 7 How many such strictly increasing sequences of integers
One quarter of the five-element subsets of {1, 2, 3, . . . , n] contain the element 7. Determine n (> 5).
Determine which row of Pascal's triangle contains three consecutive entries that are in the ratio 1 : 2 : 3.
Use the recursive technique of Example 3.9 to develop a Gray code for the 16 binary strings of length 4. Then list each of the 16 subsets of the ordered set {w, x, y, z] next to its corresponding
For positive integers n, r show that
Let A = {1, 2, 3, ... , 39, 40}. (a) Write a computer program (or develop an algorithm) to generate a random six-element subset of A. (b) For B = {2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37}, write a
Let A = {1, 2, 3, ... , 7}. Write a computer program (or develop an algorithm) that lists all the subsets B of A, where |B| = 4.
Write a computer program (or develop an algorithm) that lists all the subsets of {1, 2, 3, . . . , n), where 1 < n < 10. (The value of n should be supplied during program execution.)
Determine all of the elements in each of the following sets. (a) {1 + (-l)n| n ∈ N} (b) {n + (l/n)|n ∈ {l, 2, 3, 5, 7}} (c) {n3 + n2 | n ∈ {0, 1, 2, 3, 4}}
Consider the following six subsets of Z. A = {2m + 1| m ∈ Z} B = {2n + 3| n ∈ Z} C = {2p - 3| p ∈ Z} D = {3r + 1| r ∈ Z} E = {3s + 2| s ∈ Z} F = {3t - 2| t ∈ Z} Which of the following
For A = {1, 2, 3, 4, 5, 6, 7}, determine the number of(a) Subsets of A(b) Nonempty subsets of A(c) Proper subsets of A(d) Nonempty proper subsets of A(e) Subsets of A containing three elements(f)
(a) If a set A has 63 proper subsets, what is |A|? (b) If a set B has 64 subsets of odd cardinality, what is |B|? (c) Generalize the result of part (b)
For u = {1, 2, 3.......9, 10} let A = {1, 2, 3, 4, 5}, B = {1, 2, 4, 8}, C = {1, 2, 3, 5, 7}, and D = {2, 4, 6, 8}. Determine each of the following:(a) (A U B) © C(b) A ª (B
Write the dual statement for each of the following set- theoretic results.(a)(b) A = A © (A U B) (c) (d) A = (A U B) © (A U θ)
Let A, B ⊂ °U. Use the equivalence A ⊂ B ⇔ A ∩ B = A to show that the dual statement of A ⊂ B is the statement B ⊂ A.
Prove or disprove each of the following for sets A, B ⊂ °U. (a) P(AUB) = P(A) U P(B) (b) P(A ∩ B) = P(A) ∩ P(B)
Use membership tables to establish each of the following:(a)(b) A ª A = A (c) A ª (A © B) = A (d)
(b) How many rows are needed to construct the membership table for a set made up from the sets A1, A2, ... , An, using ©, U, and ¯?(c) Given the membership tables for two sets A, B,
Provide the justifications (selected from the laws of set theory) for the steps that are needed to simplify the set(A © B) ª [B © ((C © D) ª (C
Using the laws of set theory, simplify each of the following:(a) A © (B - A)(b) (A © B) ª (A © B © © D) ª ( ©
For each n Z+ let An = {1, 2, 3, . . . , n - 1, n}. (Here °U = Z+ and the index set I = Z+.) Determinewhere m is a fixed positive integer.
Let °U = R and let I = Z+. For each n Z+ let An = [-2n, 3n]. Determine each of the following:(a) A3(b) A4(c) A3 - A4(d) A3 A4(e)(f) (g) (h)
If A = [0, 3], B = [2, 7), with °U = R, determine each of the following: (a) A ∩ B (b) A U B (c) (d) A ∆ B (e) A - B (f) B - A
Provide the details for the proof of Theorem 3.6(b).Theorem 3.6(b)(b)
(a) Determine the sets A, B where A - B = {1,3, 7, 11}, B - A = {2, 6, 8}, and A ∩ B = {4, 9}.(b) Determine the sets C, D where C - D = {1, 2, 4}, D - C = {7, 8}, and C ∪ D = {1, 2, 4, 5, 7, 8,
Let A, B, C, D, E Z be defined as follows:A = {2n|n Z} - that is, A is the set of all (integer) multiples of 2;B = {3n|n Z}; C = {4n|n Z};D = {6n|n
Determine which of the following statements are true and which are false. (a) Z+ ⊂ Q+ (b) Z+ ⊂ Q (c) Q+ ⊂ R (d) R+ ⊂ Q (e) Q+ ∩ R+ = Q+ (f) Z+ ∪ R+ = R (g) R+ ∩ C = R+ (h) C ∪ R =
Prove each of the following results without using Venn diagrams or membership tables. (Assume a universe °U.) (a) If A ⊂ B and C ⊂ D, then A ∩ C ⊂ B ∩ D and A ∪ C ⊂ B ∪ D. (b) A ⊂
Prove or disprove each of the following: (a) For sets A, B, C ⊂ U, A ∩ C = B ∩ C ⇒ A = B. (b) For sets A, B, C ⊂ U, A U C = B U C ⇒ A = B. (c) For sets A, B, C ⊂ °U, [(A ∩ C = B ∩
Using Venn diagrams, investigate the truth or falsity of each of the following, for sets A, B, C ⊂ U. (a) A ∆ (B ∩ C) = (A ∆ B) ∩ (A ∆ C) (b) A - (B U C) = (A - B) ∩ (A - C) (c) A ∆
During freshman orientation at a small liberal arts college, two showings of the latest James Bond movie were presented. Among the 600 freshmen, 80 attended the first showing and 125 attended the
How many arrangements of the letters in CHEMIST have H before E, or E before T, or T before M? (Here "before" means anywhere before, not just immediately before.)
A manufacturer of 2000 automobile batteries is concerned about defective terminals and defective plates. If 1920 of her batteries have neither defect, 60 have defective plates, and 20 have both
A binary string of length 12 is made up of 12 bits (that is, 12 symbols, each of which is a 0 or a 1). How many such strings either start with three l's or end in four 0's?
Determine |A ∪ B ∪ C| when |A| = 50, |B| = 500, and |C| = 5000, if (a) A ⊂ B ⊂ C. (b) A ∩ 5 = A ∩ C = B ∩ C = θ. (c) |A ∩ B| = |A ∩ C| = |B ∩ C| = 3 and |A ∩ B ∩ C| = 1.
How many permutations of the 26 different letters of the alphabet contain (a) Either the pattern "OUT" or the pattern "DIG"? (b) Neither the pattern "MAN" nor the pattern "ANT"?
A six-character variable name in a certain version of ANSI FORTRAN starts with a letter of the alphabet. Each of the other five characters can be either a letter or a digit. (Repetitions are
How many arrangements of the letters in MISCELLANEOUS have no pair of consecutive identical letters?
(a) Pr(A).(b) Pr{B).(c) Pr(A © B).(d) Pr(A ª B).(e)
Twenty-five slips of paper, numbered 1, 2, 3, ... , 25, are placed in a box. If Amy draws six of these slips, without replacement, what is the probability that(a) The second smallest number drawn is
Darci rolls a fair die three times. What is the probability that(a) Her second and third rolls are both larger than her first roll?(b) The result of her second roll is greater than that of her first
At the Gamma Kappa Phi sorority the 15 sisters who are seniors line up in a random manner for a graduation picture. Two of these sisters are Columba and Piret. What is the probability that this
The freshman class of a private engineering college has 300 students. It is known that 180 can program in Java, 120 in Visual BASIC+, 30 in C++, 12 in Java and C++, 18 in Visual BASIC and C++, 12 in
An integer is selected at random from 3 through 17 inclusive. If A is the event that a number divisible by 3 is chosen and B is the event that the number exceeds 10, determine Pr(A), Pr{B), Pr(A ∩
(a) If the letters in the acronym WYSIWYG are arranged in a random manner, what is the probability the arrangement starts and ends with the same letter? (b) What is the probability that a randomly
The Tuesday night dance club is made up of six married couples and two of these twelve members must be chosen to find a dance hall for an upcoming fund raiser.(a) If the two members are selected at
If two integers are selected, at random and without replacement, from {1, 2, 3, . . ., 99, 100}, what is the probability the integers are consecutive?
Two integers are selected, at random and without replacement, from {1, 2, 3, . . . , 99, 100}. What is the probability their sum is even?
If three integers are selected, at random and without replacement, from {1, 2, 3, . . ., 99, 100}, what is the probability their sum is even?
Jerry tosses a fair coin six times. What is the probability he gets (a) All heads. (b) One head. (c) Two heads. (d) An even number of heads. (e) At least four heads?
Let if be the sample space for an experiment and let A, B be events from , where Pr(A) = 0.4, Pr(B) = 0.3, and Pr(A ˆ© B) = 0.2. Determine Pr(), Pr(), Pr(A U B), Pr(A U ), and Pr( U B).
Three types of foam are tested to see if they meet specifications. Table 3.5 summarizes the results for the 125 samples tested.Table 3.5Let A, B denote the events A: The sample has foam type 1. B:
Consider the game of Roulette as described in Example 3.44. (a) If the game is played once, what is the probability the outcome is (i) High or odd. (ii) Low or black? (b) If the game is played twice,
Let be the sample space for an experiment and let A, B Š‚ . If Pr(A) = Pr(B), Pr(A ˆ© B) = 1/5, and determine Pr(A ˆª B), Pr(A), Pr(A - B), Pr(A ˆ† B).
The nine members of a coed intramural volleyball team are to be randomly selected from nine college men and ten college women. To be classified as coed the team must include at least one player of
While traveling through Pennsylvania, Ann decides to buy a lottery ticket for which she selects seven integers from 1 to 80 inclusive. The state lottery commission then selects 11 of these 80
Let S be the sample space for an experiment and let A, B with A B. Prove that Pr(A)
Let be the sample space for an experiment and let A, B If Pr(A) = 0.7 and Pr(B) = 0.5, prove that Pr(A © B) > 0.2.
Ashley tosses a fair coin eight times. What is the probability she gets (a) Six heads. (b) At least six heads. (c) Two heads; and (d) At most two heads?
Ten ping-pong balls labeled 1 to 10 are placed in a box. Two of these balls are then drawn, in succession and without replacement, from the box. (a) Find the sample space for this experiment. (b)
Russell draws one card from a standard deck. If A, B, C denote the events A: The card is a spade. B: The card is red. C: The card is a picture card (that is, a jack, queen, or king). Find Pr(A ∪ B
A die is loaded so that the probability a given number turns up is proportional to that number. So, for example, the outcome 4 is twice as likely as the outcome 2, and the outcome 3 is three times as
Suppose we have two dice - each loaded as described in the previous exercise. If these dice are rolled, what is the probability the outcome is (a) 10. (b) At least 10. (c) A double?
Juan tosses a fair coin five times. What is the probability the number of heads always exceeds the number of tails as each outcome is observed?
Recall that in a standard deck of 52 cards there are 12 picture cards - four each of jacks, queens, and kings. Kevin draws one card from the deck. Find the probability his card is a king if we know
Alice tosses a fair coin seven times. Find the probability she gets four heads given that (a) Her first toss is a head. (b) Her first and last tosses are heads.
Paulo tosses a fair coin five times. If A, B denote the eventsA: Paulo gets an odd number of tails.B: Paulo's first toss is a tail.are A, B independent?
Paul has two coolers. The first contains eight cans of cola and three cans of lemonade. The second cooler contains five cans of cola and seven cans of lemonade. Paul randomly selects one can from the
Let W be the sample space for an experiment and let A, B, C . If events A, B are independent, events A, C are disjoint, and events B, C are independent, find Pr(B) if Pr(A) = 0.2,
An electronic system is made up of two components connected in parallel. Consequently, the system fails only when both of the components fail. The probability the first component fails is 0.05 and,
Gayla has a bag of 19 marbles of the same size. Nine of these marbles are red, six blue, and four white. She randomly selects three of the marbles, without replacement, from the bag. What is the
Let A, B, C be independent events taken from a sample space If Pr(A) = 1/8, Pr(B) = 1/4, and Pr(A U B U C) = 1/2, find Pr(C).
A company involved in the integration of personal computers gets its graphics cards from three sources. The first source provides 20% of the cards, the second source 35%, and the third source 45%.
Gustavo tosses a fair coin twice. For this experiment consider the following events: A: The first toss is a head. B : The second toss is a tail. C: The tosses result in one head and one tail. Are the
Let A, B be events taken from a sample space If Pr(A) = 0.6, Pr(B) = 0.4, and Pr(A U B) = 0.7, find Pr(A|B) and Pr(A|).
Three missiles are fired at an enemy arsenal. The probabilities the individual missiles will hit the arsenal are 0.75, 0.85, and 0.9. Find the probability that at least two of the missiles hit the
Dustin and Jennifer each toss three fair coins. What is the probability(a) Each of them gets the same number of heads?(b) Dustin gets more heads than Jennifer?(c) Jennifer gets more heads than Dustin?
Tiffany and four of her cousins play the game of "odd person out" to determine who will rake up the leaves at their grandmother Mary Lou's home. Each cousin tosses a fair coin. If the outcome for one
Ninety percent of new airport-security personnel have had prior training in weapon detection. During their first month on the job, personnel without prior training fail to detect a weapon 3% of the
The binary string 101101, where the string is unchanged upon reversing order, is called a palindrome (of length 6). Suppose a binary string of length 6 is randomly generated, with 0, 1 equally likely
In defining the notion of independence for three events we found (in Definition 3.13) that we had to check four conditions. If there are four events, say E1, E2, E3, E4, then we have to check 11
Let A, B be events taken from a sample space If Pr (A ˆ© B) = 0.1 and Pr( ˆ© ) = 0.3, what is Pr(A ˆ† B|A U B)?
Urn 1 contains 14 envelopes (of the same size) - six each contain $1 and the other eight each contain $5. Urn 2 contains eight envelopes (of the same size as those in urn 1) - three each contain $1
Let A, B be events taken from a sample space (with Pr(A) > 0 and Pr(B) > 0). If Pr(B|A)
Let A, B be events taken from a sample space If Pr(A) = 0.5, Pr(B) = 0.3, and Pr(A|B) + Pr(B|A) = 0.8, what is Pr(A © B)?
If Coach Mollet works his football team throughout August, then the probability the team will be the division champion is 0.75. The probability the coach will work his team throughout August is 0.80.
Let be the sample space for an experiment with events A, B . If Pr(A|P) = Pr(A B) = 0.5 and Pr(A U B) = 0.7, determine Pr(A) and Pr(B).
The 420 freshmen at an engineering college take either calculus or discrete mathematics (but not both). Further, both courses are offered providing either an introduction to a CAS (computer algebra
Pr(A U B) = Pr(A) + Pr()Pr(B)= Pr(B) + Pr()Pr(A).
Ceilia tosses a fair coin five times. What is the probability she gets three heads, if the first toss results in(a) A head.(b) A tail?
One bag contains 15 identical (in shape) coins - nine of silver and six of gold. A second bag contains 16 more of these coins - six silver and 10 gold. Bruno reaches in and selects one coin from the
A coin is loaded so that Pr(H) = 2/3 and Pr(T) = 1/3. Todd tosses this coin twice. Let A, B be the events A : The first toss is a tail. B : Both tosses are the same. Are A, B independent?
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