Discrete Mathematics And Its Applications 7th Edition
7th Edition
ISBN: 9781259152153
Author: Kenneth H. Rosen
Publisher: MCG CUSTOM
expand_more
expand_more
format_list_bulleted
Videos
Textbook Question
Chapter 8.2, Problem 11E
The Lucas numbers satisfy the recurrence relation
a) Show that
b) Find an explicit formula for the Lucas numbers.
Expert Solution & Answer
Want to see the full answer?
Check out a sample textbook solution
Students have asked these similar questions
Suppose that a room containing 1300 cubic feet of air is originally free of carbon monoxide (CO). Beginning
at time t = 0, cigarette smoke containing 4% CO is introduced into the room at a rate of 0.8 cubic feet per
minute. The well-circulated smoke and air mixture is allowed to leave the room at the same rate.
Let A(t) represent the amount of CO in the room (in cubic feet) after t minutes.
(A) Write the DE model for the time rate of change of CO in the room. Also state the initial condition.
dA
dt
A(0)
(B) Solve the IVP to find the amount of CO in the room at any time t > 0.
A(t)
(C) Extended exposure to a CO concentration as low as 0.00012 is harmful to the human body. Find the time
at which this concentration is reached.
t=
minutes
Newton's Law of Cooling tells us that the rate of change of the temperature of an object is proportional to
the temperature difference between the object and its surroundings. This can be modeled by the
differential equation
dT
dt
k(TA), where T is the temperature of the object after t units of time
have passed, A is the ambient temperature of the object's surroundings, and k is a constant of
proportionality.
Suppose that a cup of coffee begins at 178 degrees and, after sitting in room temperature of 61 degrees
for 12 minutes, the coffee reaches 171 degrees. How long will it take before the coffee reaches 155
degrees?
Include at least 2 decimal places in your answer.
minutes
Decide whether each limit exists. If a limit exists, estimate its
value.
11. (a) lim f(x)
x-3
f(x) ↑
4
3-
2+
(b) lim f(x)
x―0
-2
0
X
1234
Chapter 8 Solutions
Discrete Mathematics And Its Applications 7th Edition
Ch. 8.1 - Use mathematical induction to verify the formula...Ch. 8.1 - a) Find a recurrence relation for the number of...Ch. 8.1 - A vending machine dispensing books of stamps...Ch. 8.1 - A country uses as currency coins with values of 1...Ch. 8.1 - How many was are there to pay a bill of 17 pesos...Ch. 8.1 - a) Find a recurrence relation for the number of...Ch. 8.1 - a) Find a recurrence relation for the number of...Ch. 8.1 - a) Find a recurrence relation for the number of...Ch. 8.1 - a) Find a recurrence relation for the number of...Ch. 8.1 - a) Find a recurrence relation for the number of...
Ch. 8.1 - a) Find a recurrence relation for the number of...Ch. 8.1 - a) Find a recurrence relation for the number of...Ch. 8.1 - a) Find a recurrence relation for the number of...Ch. 8.1 - a) Find a recurrence relation for the number of...Ch. 8.1 - a) Find a recurrence relation for the number of...Ch. 8.1 - a) Find a recurrence relation for the number of...Ch. 8.1 - a) Find a recurrence relation for the number of...Ch. 8.1 - a) Find a recurrence relation for the number of...Ch. 8.1 - Messages are transmitted over a communications...Ch. 8.1 - A bus driver pays all tolls, using only nickels...Ch. 8.1 - a) Find the recurrence relation satisfied by Rn,...Ch. 8.1 - a) Find the recurrence relation satisfied by Rn,...Ch. 8.1 - a) Find the recurrence relation satisfied by Sn,...Ch. 8.1 - Find a recurrence relation for the number of bit...Ch. 8.1 - How many bit sequences of length seven contain an...Ch. 8.1 - a) Find a recurrence relation for the number of...Ch. 8.1 - a) Find a recurrence relation for the number of...Ch. 8.1 - Show that the Fibonacci numbers satisfy the...Ch. 8.1 - Prob. 29ECh. 8.1 - Prob. 30ECh. 8.1 - a) Use the recurrence relation developed in...Ch. 8.1 - In the Tower of Hanoi puzzle, suppose our goal is...Ch. 8.1 - Exercises 33-37 deal with a variation of the...Ch. 8.1 - Exercises 33-37 deal with a variation of the...Ch. 8.1 - Prob. 35ECh. 8.1 - Exercises 33-37 deal with a variation of the...Ch. 8.1 - Prob. 37ECh. 8.1 - Prob. 38ECh. 8.1 - Show that the Reve’s puzzle with four disks can be...Ch. 8.1 - Prob. 40ECh. 8.1 - Show that if R(n) is the number of moves used by...Ch. 8.1 - Prob. 42ECh. 8.1 - Prob. 43ECh. 8.1 - Prob. 44ECh. 8.1 - Prob. 45ECh. 8.1 - Prob. 46ECh. 8.1 - Prob. 47ECh. 8.1 - Prob. 48ECh. 8.1 - Show that an2=an2an+2an .Ch. 8.1 - Prob. 50ECh. 8.1 - Prob. 51ECh. 8.1 - Prob. 52ECh. 8.1 - Construct the algorithm described in the text...Ch. 8.1 - Use Algorithm 1 to determine the maximum number of...Ch. 8.1 - For each part of Exercise 54, use your algorithm...Ch. 8.1 - In this exercise we will develop a dynamic...Ch. 8.1 - Dynamic programming can be used to develop an...Ch. 8.2 - Determine which of these are linear homogeneous...Ch. 8.2 - Determine which of these are linear homogeneous...Ch. 8.2 - Solve these recurrence relations together with the...Ch. 8.2 - Solve these recurrence relations together with the...Ch. 8.2 - Prob. 5ECh. 8.2 - Prob. 6ECh. 8.2 - Prob. 7ECh. 8.2 - A model for the number of lobsters caught per year...Ch. 8.2 - Prob. 9ECh. 8.2 - Prob. 10ECh. 8.2 - The Lucas numbers satisfy the recurrence relation...Ch. 8.2 - Find the solution to an=2an1+an2+2an3 for n = 3,4,...Ch. 8.2 - Find the solution to an=7an2+6an3 with a0=9,a1=10...Ch. 8.2 - Find the solution to an=5an24an4 with...Ch. 8.2 - Prob. 15ECh. 8.2 - Prob. 16ECh. 8.2 - Prove this identity relating the Fibonacci numbers...Ch. 8.2 - Solve the recurrence relation an=6an112an2+8an3...Ch. 8.2 - Prob. 19ECh. 8.2 - Prob. 20ECh. 8.2 - Prob. 21ECh. 8.2 - What is the general form of the solutions of a...Ch. 8.2 - Consider the nonhomogeneous linear recurrence...Ch. 8.2 - Consider the nonhomogeneous linear recurrence...Ch. 8.2 - a) Determine values of the constants A and B such...Ch. 8.2 - What is the general form of the particular...Ch. 8.2 - What is the general form of the particular...Ch. 8.2 - a) Find all solutions of the recurrence relation...Ch. 8.2 - Prob. 29ECh. 8.2 - Prob. 30ECh. 8.2 - Find all solutions of the recurrence relation...Ch. 8.2 - Find the solution of the recurrence relation...Ch. 8.2 - Prob. 33ECh. 8.2 - Prob. 34ECh. 8.2 - Find the solution of the recurrence relation...Ch. 8.2 - Prob. 36ECh. 8.2 - Prob. 37ECh. 8.2 - Prob. 38ECh. 8.2 - Prob. 39ECh. 8.2 - Solve the simultaneous recurrence relations...Ch. 8.2 - Prob. 41ECh. 8.2 - Prob. 42ECh. 8.2 - Prob. 43ECh. 8.2 - Prob. 44ECh. 8.2 - Prob. 45ECh. 8.2 - Suppose that there are two goats on an island...Ch. 8.2 - Prob. 47ECh. 8.2 - Prob. 48ECh. 8.2 - Use Exercise 48 to solve the recurrence relation...Ch. 8.2 - It can be shown that Cn, the average number of...Ch. 8.2 - Prob. 51ECh. 8.2 - Prob. 52ECh. 8.2 - Prob. 53ECh. 8.3 - How many comparisons are needed for a binary...Ch. 8.3 - Prob. 2ECh. 8.3 - Multiply (1110)2 and (1010)2 using the fast...Ch. 8.3 - Express the fast multiplication algorithm in...Ch. 8.3 - Determine a value for the constant C in Example...Ch. 8.3 - Prob. 6ECh. 8.3 - Prob. 7ECh. 8.3 - Suppose that f(n)=2f(n/2)+3 when is an even...Ch. 8.3 - Prob. 9ECh. 8.3 - Find f(n) when n=2k , where f satisfies the...Ch. 8.3 - Give a big-O estimate for the function f in...Ch. 8.3 - Find f(n) when n=3k , where f satisfies the...Ch. 8.3 - Give a big-O estimate for the function f in...Ch. 8.3 - Suppose that there are n=2k terms in an...Ch. 8.3 - How many rounds are in the elimination tournament...Ch. 8.3 - Prob. 16ECh. 8.3 - Suppose that the votes of n people for different...Ch. 8.3 - Suppose that each person in a group of n people...Ch. 8.3 - a) Set up a divide-and-conquer recurrence relation...Ch. 8.3 - a) Set up a divide-and-conquer recurrence relation...Ch. 8.3 - Suppose that the function f satisfies the...Ch. 8.3 - Suppose that the function f satisfies the...Ch. 8.3 - This exercise deals with the problem of finding...Ch. 8.3 - Apply the algorithm described in Example 12 for...Ch. 8.3 - Apply the algorithm described in Example 12 for...Ch. 8.3 - Use pseudocode to describe the recursive algorithm...Ch. 8.3 - Prob. 27ECh. 8.3 - Prob. 28ECh. 8.3 - In Exercises 29-33, assume that f is an increasing...Ch. 8.3 - Prob. 30ECh. 8.3 - Prob. 31ECh. 8.3 - Prob. 32ECh. 8.3 - Prob. 33ECh. 8.3 - In Exercises 29-33, assume that f is an increasing...Ch. 8.3 - In Exercises 29-33, assume that f is an increasing...Ch. 8.3 - In Exercises 29-33, assume that f is an increasing...Ch. 8.3 - In Exercises 29-33, assume that f is an increasing...Ch. 8.4 - Find the generating function for the finite...Ch. 8.4 - Find the generating function for the finite...Ch. 8.4 - In Exercises 3-8, by a closed form we mean an...Ch. 8.4 - In Exercises 3-8, by a closed form we mean an...Ch. 8.4 - Prob. 5ECh. 8.4 - In Exercises 3-8, by a closed form we mean an...Ch. 8.4 - In Exercises 3-8, by a closed form we mean an...Ch. 8.4 - In Exercises 3-8, by a closed form we mean an...Ch. 8.4 - Find the coefficient of x10in the power series of...Ch. 8.4 - Prob. 10ECh. 8.4 - Prob. 11ECh. 8.4 - Prob. 12ECh. 8.4 - Use generating functions to determine the number...Ch. 8.4 - Use generating functions to determine the number...Ch. 8.4 - Use generating functions to determine the number...Ch. 8.4 - Use generating functions to find the number of...Ch. 8.4 - In how many ways can 25 identical donuts be...Ch. 8.4 - Use generating functions to find the number of...Ch. 8.4 - Prob. 19ECh. 8.4 - Prob. 20ECh. 8.4 - Prob. 21ECh. 8.4 - Prob. 22ECh. 8.4 - Prob. 23ECh. 8.4 - Prob. 24ECh. 8.4 - Explain how generating functions can be used to...Ch. 8.4 - Explain how generating functions can be used to...Ch. 8.4 - Prob. 27ECh. 8.4 - Prob. 28ECh. 8.4 - Use generating functions (and a computer algebra...Ch. 8.4 - Use generating functions (and a computer algebra...Ch. 8.4 - Prob. 31ECh. 8.4 - If G(x) is the generating function for the...Ch. 8.4 - Prob. 33ECh. 8.4 - Prob. 34ECh. 8.4 - Prob. 35ECh. 8.4 - Use generating functions to solve the recurrence...Ch. 8.4 - Prob. 37ECh. 8.4 - Use generating functions to solve the recurrence...Ch. 8.4 - Use generating functions to solve the recurrence...Ch. 8.4 - Prob. 40ECh. 8.4 - Prob. 41ECh. 8.4 - Prob. 42ECh. 8.4 - (Calculus required) Let {Cn}be the sequence of...Ch. 8.4 - Use generating functions to prove Pascal’s...Ch. 8.4 - Use generating functions to prove Vandermonde’s...Ch. 8.4 - Prob. 46ECh. 8.4 - Prob. 47ECh. 8.4 - Prob. 48ECh. 8.4 - Find the sequence with each of these functions as...Ch. 8.4 - Find the sequence with each of these functions as...Ch. 8.4 - A coding system encodes messages using strings of...Ch. 8.4 - A coding system encodes messages using strings of...Ch. 8.4 - Generating functions are useful in studying the...Ch. 8.4 - Generating functions are useful in studying the...Ch. 8.4 - Prob. 55ECh. 8.4 - Prob. 56ECh. 8.4 - Generating functions are useful in studying the...Ch. 8.4 - Generating functions are useful in studying the...Ch. 8.4 - Suppose that X is a random variable on a sample...Ch. 8.4 - Prob. 60ECh. 8.4 - Prob. 61ECh. 8.4 - Show that if X and Y are independent random...Ch. 8.5 - How many elements are in A1A2 if there are 12...Ch. 8.5 - There are 345 students at a college who have taken...Ch. 8.5 - A survey of households in the United States...Ch. 8.5 - A marketing report concerning personal computers...Ch. 8.5 - Find the number of elements A1A2A3 if there are...Ch. 8.5 - Prob. 6ECh. 8.5 - There are 2504 computer science students at a...Ch. 8.5 - In a survey of 270 college students, it is found...Ch. 8.5 - How many students are enrolled in a course either...Ch. 8.5 - Find the number of positive integers not exceeding...Ch. 8.5 - Find the number of positive integers not exceeding...Ch. 8.5 - Find the number of positive integers not exceeding...Ch. 8.5 - Find the number of positive integers not exceeding...Ch. 8.5 - Find the number of positive integers not exceeding...Ch. 8.5 - How many swings of length eight do not contain six...Ch. 8.5 - How many permutations of the 26 letters of the...Ch. 8.5 - How many permutations of the 10 digits either...Ch. 8.5 - Prob. 18ECh. 8.5 - Prob. 19ECh. 8.5 - How many terms are there in the formula for the...Ch. 8.5 - Prob. 21ECh. 8.5 - Prob. 22ECh. 8.5 - Prob. 23ECh. 8.5 - Prob. 24ECh. 8.5 - Let E1, E2 ,and E3 be three events from a sample...Ch. 8.5 - Prob. 26ECh. 8.5 - Find the probability that when four numbers from 1...Ch. 8.5 - Prob. 28ECh. 8.5 - Prob. 29ECh. 8.5 - Prob. 30ECh. 8.5 - Prob. 31ECh. 8.6 - Suppose that in a bushel of 100 apples there are...Ch. 8.6 - Prob. 2ECh. 8.6 - Prob. 3ECh. 8.6 - Prob. 4ECh. 8.6 - Find the number of primes less than 200 using the...Ch. 8.6 - Prob. 6ECh. 8.6 - How many positive integers less than 10,000 are...Ch. 8.6 - Prob. 8ECh. 8.6 - How many ways are there to distribute six...Ch. 8.6 - In how many ways can eight distinct balls be...Ch. 8.6 - In how many ways can seven different jobs be...Ch. 8.6 - List all the derangements of {1, 2,3, 4}.Ch. 8.6 - Prob. 13ECh. 8.6 - Prob. 14ECh. 8.6 - A machine that inserts letters into envelopes goes...Ch. 8.6 - A group of n students is assigned seats for each...Ch. 8.6 - Prob. 17ECh. 8.6 - Prob. 18ECh. 8.6 - Prob. 19ECh. 8.6 - Prob. 20ECh. 8.6 - For which positive integers n is Dn, the number of...Ch. 8.6 - Prob. 22ECh. 8.6 - Prob. 23ECh. 8.6 - Prob. 24ECh. 8.6 - Prob. 25ECh. 8.6 - How many derangements of {1, 2, 3, 4, 5, 6} end...Ch. 8.6 - Prove Theorem 1.Ch. 8 - a) What is a recurrence re1aon? b) Find a...Ch. 8 - Explain how the Fibonacci numbers are used to...Ch. 8 - a) Find a recurrence relation for the number of...Ch. 8 - Prob. 6RQCh. 8 - a) Explain how to solve linear homogeneous...Ch. 8 - Prob. 8RQCh. 8 - Prob. 9RQCh. 8 - a) Give a formula for the number of elements in...Ch. 8 - a) Give a formula for the number of elements in...Ch. 8 - Prob. 12RQCh. 8 - Explain how the principle of inclusion-exclusion...Ch. 8 - Prob. 14RQCh. 8 - Prob. 15RQCh. 8 - a) Define a derangement. b) Why is counting the...Ch. 8 - A group of 10 people begin a chain letter, with...Ch. 8 - A nuclear reactor has created 18 grams of a...Ch. 8 - Every hour the U.S. government prints 10,000 more...Ch. 8 - Suppose that every hour there are two new bacteria...Ch. 8 - Messages are sent over a communications channel...Ch. 8 - Prob. 6SECh. 8 - How many ways are there to form these postages...Ch. 8 - Prob. 8SECh. 8 - Solve the recurrence relation an=a2n1/bn2 if a0=1...Ch. 8 - Prob. 10SECh. 8 - Find the solution of the recurrence relation...Ch. 8 - Prob. 12SECh. 8 - Prob. 13SECh. 8 - Prob. 14SECh. 8 - Prob. 15SECh. 8 - In Exercises 15-18 we develop a dynamic...Ch. 8 - In Exercises 15-18 we develop a dynamic...Ch. 8 - In Exercises 15-18 we develop a dynamic...Ch. 8 - Find the solution to the recurrence relation...Ch. 8 - Find the solution to the recurrence relation...Ch. 8 - Give a big-O estimate for the size of f in...Ch. 8 - Find a recurrence relation that describes the...Ch. 8 - Prob. 23SECh. 8 - Prob. 24SECh. 8 - Prob. 25SECh. 8 - Find an where a) an=3 . b) an=4n+7 . c) an=n2+n+1Ch. 8 - Prob. 27SECh. 8 - Prob. 28SECh. 8 - Prob. 29SECh. 8 - Prob. 30SECh. 8 - Prob. 31SECh. 8 - Prob. 32SECh. 8 - Prob. 33SECh. 8 - Prob. 34SECh. 8 - Prob. 35SECh. 8 - How many terms are needed when the...Ch. 8 - How many solutions in positive integers are there...Ch. 8 - How many positive integers less than 1,000,000 are...Ch. 8 - How many positive integers less than 200 are a)...Ch. 8 - How many ways are there to assign six different...Ch. 8 - What is the probability that exactly one person is...Ch. 8 - How many bit stings of length six do not contain...Ch. 8 - What is the probability that a bit string of...Ch. 8 - Prob. 1CPCh. 8 - Prob. 2CPCh. 8 - Prob. 3CPCh. 8 - Prob. 4CPCh. 8 - Prob. 5CPCh. 8 - Prob. 6CPCh. 8 - Prob. 7CPCh. 8 - Prob. 8CPCh. 8 - Prob. 9CPCh. 8 - Prob. 10CPCh. 8 - Prob. 11CPCh. 8 - Prob. 12CPCh. 8 - Given a positive integer n, list all the...Ch. 8 - Prob. 1CAECh. 8 - Prob. 2CAECh. 8 - Find as many prime Fibonacci numbers as you can....Ch. 8 - Prob. 4CAECh. 8 - Prob. 5CAECh. 8 - Prob. 6CAECh. 8 - Prob. 7CAECh. 8 - Prob. 8CAECh. 8 - Prob. 9CAECh. 8 - List all the derangements of 1,2,3,4,5,6,7,8 .Ch. 8 - Prob. 11CAECh. 8 - Find the original source where Fibonacci presented...Ch. 8 - Explain how the Fibonacci numbers arise in a...Ch. 8 - Prob. 3WPCh. 8 - Discuss as mans different problems as possible...Ch. 8 - Prob. 5WPCh. 8 - Prob. 6WPCh. 8 - Prob. 7WPCh. 8 - Prob. 8WPCh. 8 - Describe the solution of Ulam’s problem (see...Ch. 8 - Discuss variations of Ulam’s problem (see Exercise...Ch. 8 - Prob. 11WPCh. 8 - Describe how sieve methods are used in number...Ch. 8 - Look up the rules of the old French card game of...Ch. 8 - Prob. 14WPCh. 8 - Describe the Polyá theory of counting and the kind...Ch. 8 - The problème des ménages (the problem of the...Ch. 8 - Explain how rook polynomials can be used to solve...
Knowledge Booster
Learn more about
Need a deep-dive on the concept behind this application? Look no further. Learn more about this topic, subject and related others by exploring similar questions and additional content below.Similar questions
- can you help me solve this question and show workings pleasearrow_forwardhow could the bar graph have been organized differently to make it easier to compare opinion changes within political partiesarrow_forwardketch a graph of the function f(x) = 3 cos (표) 6. x +1 5 4 3 3 80 9 2+ 1 -9 -8 -7 -6 -5 -4 -3 -2 -1 1 2 3 4 5 6 7 -1 -2 -3+ -4 5 -6+ Clear All Draw: пи > Next Questionarrow_forward
- Draw the following graph on the interval πT 5π < x < 2 2 y = 2 sin (2(x+7)) 6. 5. 4 3 3 2 1 +3 /2 -π/3 -π/6 π/6 π/3 π/2 2π/3 5π/6 π 7π/6 4π/3 3π/2 5π/311π/6 2π 13π/67π/3 5π Clear All Draw:arrow_forwardLet f : X → Y and g : Y → Z be two functions. Prove that(1) if g ◦ f is injective, then f is injective; (2) if g ◦ f is surjective, then g is surjective.arrow_forwardketch a graph of the function f(x) = 3 cos (표) 6. x +1 5 4 3 3 80 9 2+ 1 -9 -8 -7 -6 -5 -4 -3 -2 -1 1 2 3 4 5 6 7 -1 -2 -3+ -4 5 -6+ Clear All Draw: пи > Next Questionarrow_forward
- Solve the following boundary value problem using method of separation of variables ди 11.07 (137) 1 J²u + = = 0, -Пarrow_forward3 2 20-10-18-17-16-15-14-13-12-11-10-9 -8 -7 -6 -$4-3-2-1 -1 -2 -3 4- -5+ The curve above is the graph of a sinusoidal function. It goes through the points (-8, -4) and (6,-4). Find a sinusoidal function that matches the given graph. If needed, you can enter π=3.1416... as 'pi' in your answer, otherwise use at least 3 decimal digits. f(x) = > Next Question Barrow_forwardX Grades for X Assignmen X A-Z Datab XE Biocultural X EBSCO-Ful X Review es/119676/assignments/3681238 Review Quiz 8.1-p2 points possible Answered: 3/5 ● Question 1 4+ 3. 2 1 13 /12-11 -10 -9 -8 -7 -6 -5 -4 -3 -2 -1 1 2 3 4 -1 -2 -3 -4- 5 2 6 The curve above is the graph of a sinusoidal function. It goes through the points (-7,0) and (3,0). Find a sinusoidal function that matches the given graph. If needed, you can enter π=3.1416... as 'pi' in your answer, otherwise use at least 3 decimal digits. f(x) = > Next Question 申 J % F 刀 Q Search S € t ח Y 7 I * 00 J ப I Darrow_forward2 d) Draw the following graph on the interval k 5π Next Questionarrow_forwardDraw the following graph on the interval 5л Next Questionarrow_forwardDetermine whether the lines L₁ (t) = (-2,3, −1)t + (0,2,-3) and L2 p(s) = (2, −3, 1)s + (-10, 17, -8) intersect. If they do, find the point of intersection.arrow_forwardarrow_back_iosSEE MORE QUESTIONSarrow_forward_ios
Recommended textbooks for you
Algebra & Trigonometry with Analytic GeometryAlgebraISBN:9781133382119Author:SwokowskiPublisher:Cengage
Linear Algebra: A Modern IntroductionAlgebraISBN:9781285463247Author:David PoolePublisher:Cengage Learning
College Algebra (MindTap Course List)AlgebraISBN:9781305652231Author:R. David Gustafson, Jeff HughesPublisher:Cengage Learning
Algebra & Trigonometry with Analytic Geometry
Algebra
ISBN:9781133382119
Author:Swokowski
Publisher:Cengage
Linear Algebra: A Modern Introduction
Algebra
ISBN:9781285463247
Author:David Poole
Publisher:Cengage Learning
College Algebra (MindTap Course List)
Algebra
ISBN:9781305652231
Author:R. David Gustafson, Jeff Hughes
Publisher:Cengage Learning
What is a Relation? | Don't Memorise; Author: Don't Memorise;https://www.youtube.com/watch?v=hV1_wvsdJCE;License: Standard YouTube License, CC-BY
RELATIONS-DOMAIN, RANGE AND CO-DOMAIN (RELATIONS AND FUNCTIONS CBSE/ ISC MATHS); Author: Neha Agrawal Mathematically Inclined;https://www.youtube.com/watch?v=u4IQh46VoU4;License: Standard YouTube License, CC-BY