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mathematics
statistics

Questions and Answers of Statistics

Use a Venn diagram to illustrate the set of all months of the year whose names do not contain the letter R in the set of all months of the year.
Use a Venn diagram to illustrate the relationships A ⊂ B and B ⊂ C.
Suppose that A, B, and C are sets such that A ⊆ B and B ⊆ C. Show that A ⊆ C.
What is the cardinality of each of these sets? a) {a} b) {{a}} c) {a, {a}} d) {a, {a}, {a, {a}}}
Find the power set of each of these sets, where a and b are distinct elements. a) {a} b) {a, b} c) {∅, {∅}}
How many elements does each of these sets have where a and b are distinct elements? a) P({a, b, {a, b}}) b) P({∅, a, {a}, {{a}}}) c) P(P(∅))
Prove that P(A) ⊆ P(B) if and only if A ⊆ B.
Let A = {a, b, c, d} and B = {y, z}. Find a) A × B. b) B × A.
What is the Cartesian product A × B × C, where A is the set of all airlines and B and C are both the set of all cities in the United States? Give an example of how this Cartesian product can be
For each of these pairs of sets, determine whether the first is a subset of the second, the second is a subset of the first, or neither is a subset of the other. a) The set of airline flights from
Let A be a set. Show that ∅×A = A×∅ = ∅.
Find A2 if a) A = {0, 1, 3}. b) A = {1, 2, a, b}.
How many different elements does A × B have if A has m elements and B has n elements?
How many different elements does An have when A has m elements and n is a positive integer?
Explain why A × B × C and (A × B) × C are not the same.
Translate each of these quantifications into English and determine its truth value. a) ∀x∈R (x2 ≠ −1) b) ∃x∈Z (x2 = 2) c) ∀x∈Z (x2 > 0) d) ∃x∈R (x2 = x)
Find the truth set of each of these predicates where the domain is the set of integers.a) P(x): x2 < 3b) Q(x): x2 > xc) R(x): 2x + 1 = 0
The defining property of an ordered pair is that two ordered pairs are equal if and only if their first elements are equal and their second elements are equal. Surprisingly, instead of taking the
Describe a procedure for listing all the subsets of a finite set.
Determine whether each of these pairs of sets are equal. a) {1, 3, 3, 3, 5, 5, 5, 5, 5}, {5, 3, 1} b) {{1}}, {1, {1}} c) ∅, {∅}
For each of the following sets, determine whether 2 is an element of that set. a) {x ∈ R | x is an integer greater than 1} b) {x ∈ R | x is the square of an integer} c) {2, {2}} d) {{2},
Determine whether each of these statements is true or false. a) 0 ∈ ∅ b) ∅ ∈ {0} c) {0} ⊂ ∅ d) ∅ ⊂ {0} e) {0} ∈ {0} f) {0} ⊂ {0} g) {∅} ⊆ {∅}
Let A be the set of students who live within one mile of school and let B be the set of students who walk to classes. Describe the students in each of these sets. a) A ∩ B b) A ∪ B c) A - B d) B
Let A and B be sets. Prove the commutative laws from Table 1 by showing thata) A ∪ B = B ∪ A.b) A ∩ B = B ∩ A.
Prove the second absorption law from Table 1 by showing that if A and B are sets, then A ∩ (A ∪ B) = A.
Prove the first associative law from Table 1 by showing that if A, B, and C are sets, then A ∪ (B ∪ C) = (A ∪ B) ∪ C.
Prove the first distributive law from Table 1 by showing that if A, B, and C are sets, then A ∪ (B ∩ C) = (A ∪ B) ∩ (A ∪ C).
Let A = {0, 2, 4, 6, 8, 10}, B = {0, 1, 2, 3, 4, 5, 6}, and C = {4, 5, 6, 7, 8, 9, 10}. Find a) A ∩ B ∩ C. b) A ∪ B ∪ C. c) (A ∪ B) ∩ C. d) (A ∩ B) ∪ C.
What can you say about the sets A and B if we know that a) A ∪ B = A? b) A ∩ B = A? c) A − B = A? d) A ∩ B = B ∩ A? e) A − B = B − A?
Let A = {a, b, c, d, e} and B = {a, b, c, d, e, f, g, h}. Find a) A ∪ B. b) A ∩ B. c) A − B. d) B − A.
Show that A ⊕ B = (A ∪ B) − (A ∩ B).
Show that if A is a subset of a universal set U, then a) A ⊕ A = ∅. b) A⊕∅ = A. c) A ⊕ U = A. d) A ⊕ A = U.
What can you say about the sets A and B if A ⊕ B = A?
Suppose that A, B, and C are sets such that A ⊕ C = B ⊕ C. Must it be the case that A = B?
Show that if A is an infinite set, then whenever B is a set, A ∪ B is also an infinite set.
Let Ai = {1, 2, 3, . . . , i} for i = 1, 2, 3, . . . . Find(a)(b)
Let Ai be the set of all nonempty bit strings (that is, bit strings of length at least one) of length not exceeding i. Find(a)(b)
Using the same universal set as in the last problem, find the set specified by each of these bit strings. a) 11 1100 1111 b) 01 0111 1000 c) 10 0000 0001
What is the bit string corresponding to the difference of two sets?
Show how bitwise operations on bit strings can be used to find these combinations of A = {a, b, c, d, e}, B = {b, c, d, g, p, t, v}, C = {c, e, i, o, u, x, y, z}, and D = {d, e, h, i, n, o, t, u, x,
Find the successors of the following sets. a) {1, 2, 3} b) ∅ c) {∅} d) {∅, {∅}}
Let A and B be the multisets {3 · a, 2 · b, 1 · c} and {2 · a, 3 · b, 4 · d}, respectively. Find a) A ∪ B. b) A ∩ B. c) A − B. d) B − A. e) A + B.
The complement of a fuzzy set S is the set , with the degree of the membership of an element in equal to 1 minus the degree of membership of this element in S. Find (the fuzzy set of people who
Prove the domination laws in Table 1 by showing thata) A ∪ U = U.b) A∩∅ = ∅.
Prove the complement laws in Table 1 by showing that a) A ∪ = U. b) A ∩ = ∅.
Find the domain and range of these functions. Note that in each case, to find the domain, determine the set of elements assigned values by the function. a) The function that assigns to each bit
a) Prove that a strictly decreasing function from R to itself is one-to-one.b) Give an example of a decreasing function from R to itself that is not one-to-one.
Let f (x) = x2/3. Find f (S) if a) S = {−2,−1, 0, 1, 2, 3}. b) S = {0, 1, 2, 3, 4, 5}. c) S = {1, 5, 7, 11}. d) S = {2, 6, 10, 14}.
Show that the function f (x) = |x| from the set of real numbers to the set of nonnegative real numbers is not invertible, but if the domain is restricted to the set of nonnegative real numbers, the
Suppose that g is a function from A to B and f is a function from B to C. a) Show that if both f and g are one-to-one functions, then f ◦ g is also one-to-one. b) Show that if both f and g are onto
If f and f ◦ g are onto, does it follow that g is onto? Justify your answer.
Find f + g and fg for the functions f and g given in Exercise 36.
Show that the function f (x) = ax + b from R to R is invertible, where a and b are constants, with a ≠ 0, and find the inverse of f.
Let g(x) = x. Find a) g −1 ({0}). b) g −1({−1, 0, 1}). c) g −1 ({x | 0 < x < 1}).
Show that if x is a real number and n is an integer, then a) x < n if and only if x< n. b) n < x if and only if n < x.
The function INT is found on some calculators, where INT(x) = xwhen x is a nonnegative real number and INT(x) = xwhen x is a negative real number. Show that this INT function satisfies the identity
Why is f not a function from R to R if a) f (x) = 1/x? b) f (x) =√x? c) f (x) = ± √(x2 + 1)?
Let a and b be real numbers with a < b. Use the floor and/or ceiling functions to express the number of integers n that satisfy the inequality a < n < b.
How many bytes are required to encode n bits of data where n equals? a) 7? b) 17? c) 1001? d) 28,800?
Data are transmitted over a particular Ethernet network in blocks of 1500 octets (blocks of 8 bits). How many blocks are required to transmit the following amounts of data over this Ethernet network?
Draw the graph of the function f (x) = 2xfrom R to R.
Drawthe graph of the functionf (x) = x + x/2from R to R.
Determine whether f is a function from the set of all bit strings to the set of integers ifa) f (S) is the position of a 0 bit in S.b) f (S) is the number of 1 bits in S.c) f (S) is the smallest
Which functions in Exercise 12 are onto?In exercise 12a) f (n) = n − 1 b) f (n) = n2 + 1c) f (n) = n3 d) f (n) = n/2
Determine whether the function f : Z × Z → Z is onto if a) f (m, n) = m + n. b) f (m, n) = m2 + n2. c) f (m, n) = m. d) f (m, n) = |n|. e) f (m, n) = m − n.
Consider these functions from the set of teachers in a school. Under what conditions is the function one-to-one if it assigns to a teacher his or her a) Office. b) Assigned bus to chaperone in a
Specify a codomain for each of the functions in Exercise 17. Under what conditions is each of the functions with the codomain you specified onto? In exercise 17 a) Office. b) Assigned bus to
Determine whether each of these functions is a bijection from R to R. a) f (x) = 2x + 1 b) f (x) = x2 + 1 c) f (x) = x3 d) f (x) = (x2 + 1)/(x2 + 2)
Let f : R → R and let f (x) > 0 for all x ∈ R. Show that f (x) is strictly decreasing if and only if the function g(x) = 1/f (x) is strictly increasing.
List all the steps used byAlgorithm 1 to find the maximum of the list 1, 8, 12, 9, 11, 2, 14, 5, 10, 4.
Describe an algorithm that interchanges the values of the variables x and y, using only assignments. What is the minimum number of assignment statements needed to do this?
List all the steps used to search for 9 in the sequence 1,3, 4, 5, 6, 8, 9, 11 usinga) A linear search. b) A binary search.
Describe an algorithm that inserts an integer x in the appropriate position into the list a1, a2, . . . , an of integers that are in increasing order.
Describe an algorithm that locates the first occurrence of the largest element in a finite list of integers, where the integers in the list are not necessarily distinct.
Describe an algorithm that produces the maximum, median, mean, and minimum of a set of three integers. (The median of a set of integers is the middle element in the list when these integers are
Describe an algorithm that puts the first three terms of a sequence of integers of arbitrary length in increasing order.
Describe an algorithm that determines whether a function from a finite set of integers to another finite set of integers is onto.
Describe an algorithm that will count the number of 1s in a bit string by examining each bit of the string to determine whether it is a 1 bit.
The ternary search algorithm locates an element in a list of increasing integers by successively splitting the list into three sublists of equal (or as close to equal as possible) size, and
Devise an algorithm that finds a mode in a list of nondecreasing integers. (Recall that a list of integers is nondecreasing if each term is at least as large as the preceding term.)
Devise an algorithm that finds the sum of all the integers in a list.
Devise an algorithm that finds the first term of a sequence of integers that equals some previous term in the sequence.
Devise an algorithm that finds the first term of a sequence of positive integers that is less than the immediately preceding term of the sequence.
Use the bubble sort to sort 3, 1, 5, 7, 4, showing the lists obtained at each step.
Adapt the bubble sort algorithm so that it stops when no interchanges are required. Express this more efficient version of the algorithm in pseudocode.
Use the insertion sort to sort the list in Exercise 35, showing the lists obtained at each step.
Sort these lists using the selection sort. a) 3, 5, 4, 1, 2 b) 5, 4, 3, 2, 1 c) 1, 2, 3, 4, 5
Describe an algorithm based on the linear search for determining the correct position in which to insert a new element in an already sorted list.
How many comparisons does the insertion sort use to sort the list 1, 2, . . . , n?
Show all the steps used by the binary insertion sort to sort the list 3, 2, 4, 5, 1, 6.
Express the binary insertion sort in pseudocode.
Describe an algorithm that takes as input a list of n integers in nondecreasing order and produces the list of all values that occur more than once. (Recall that a list of integers is nondecreasing
When a list of elements is in close to the correct order, would it be better to use an insertion sort or its variation described in Exercise 50?
Use the greedy algorithm to make change using quarters, dimes, nickels, and pennies fora) 51 cents. b) 69 cents.c) 76 cents. d) 60 cents.
Use the greedy algorithm to make change using quarters, dimes, and pennies (but no nickels) for each of the amounts given in Exercise 53. For which of these amounts does the greedy algorithm use the
Use Algorithm 7 to schedule the largest number of talks in a lecture hall from a proposed set of talks, if the starting and ending times of the talks are 9:00 a.m. and 9:45 a.m.; 9:30 a.m. and 10:00
a) Devise a greedy algorithm that determines the fewest lecture halls needed to accommodate n talks given the starting and ending time for each talk.b) Prove that your algorithm is optimal. Suppose
Write the deferred acceptance algorithm in pseudocode.
Showthat the deferred acceptance always terminates with a stable assignment.
Show that the following problem is solvable. Given two programs with their inputs and the knowledge that exactly one of them halts, determine which halts.

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