5th Edition

ISBN: 9780357035283

Author: EPP

Publisher: Cengage

See similar textbooks

Concept explainers

Statements and Logic

Logic is the analysis of general patterns of thoughts without regard to any specific sense of context.

Videos

Textbook Question

Chapter 7.1, Problem 51ES

Each of exercises 51-53 refers to the Euler phi function, denoted ϕ , which is defined as follows: For each integer n ≥ 1 , ϕ ( n ) is the number of positive integers less than or equal to n that have no common factors with n except ± 1 . For example, ϕ ( 10 ) = 4 because there are four positive integers less than or equal to 10 that have no common factor with 10 except ± 1 -namely, 1,3,7, and9.

Find each of the followinga

a. ϕ (15)
b. ϕ (2)
c. ϕ (5)
d. ϕ (12)
e. ϕ (11)
e. ϕ (1)

Expert Solution & Answer

Want to see the full answer?

Check out a sample textbook solution

See solution

Chapter 7.1, Problem 50ES

Chapter 7.1, Problem 52ES

Students have asked these similar questions

Examples: Solve the following differential equation using Laplace transform (e) ty"-ty+y=0 with y(0) = 0, and y'(0) = 1

Examples: Solve the following differential equation using Laplace transform (a) y" +2y+y=t with y(0) = 0, and y'(0) = 1

Temperature for Sudbury (degrees Celsius) 3. The following table gives the mean monthly temperatures for Sudbury, Ontario and Windsor, Ontario. Each month is represented by the day of the year in the middle of the month. Month Day of Year Temperature for Windsor (degrees Celsius) January 15 -13.7 -4.7 February 45 -11.9 -3.8 March 75 -5.9 2.3 April 106 3.0 8.7 May 136 10.6 14.6 June 167 15.8 20.2 July 197 18.9 22.6 August 228 17.4 22.0 September 259 12.2 17.9 October 289 6.2 11.5 November 320 -1.2 4.8 December 350 -10.1 -1.2 a) Create a scatter plot of temperature vs. day of the year for each city. b) Draw the curve of best fit for each graph. c) Use your graphs to estimate when the temperature increases fastest, for each set of temperature data. Explain how you determined these values. d) Use your graphs to estimate the rate at which the temperature is increasing at the two times from question 3. e) Determine an equation of a sinusoidal function to model the data for each city

Chapter 7 Solutions

Discrete Mathematics With Applications

Ch. 7.1 - Given a function f from a set X to a set Y, f(x)...Ch. 7.1 - Given a function f from a set X to a set Y, if...Ch. 7.1 - Prob. 3TY Ch. 7.1 - Given a function f then a set X to a set Y, if...Ch. 7.1 - Prob. 5TY Ch. 7.1 - Prob. 6TY Ch. 7.1 - Prob. 7TY Ch. 7.1 - Prob. 8TY Ch. 7.1 - Prob. 9TY Ch. 7.1 - Prob. 1ES

Ch. 7.1 - Let X={1,3,5} and Y={a,b,c,d}. Define g:XY by the...Ch. 7.1 - Indicate whether the statement in parts (a)-(d)...Ch. 7.1 - a. Find all function from X={a,b}toY={u,v} . b....Ch. 7.1 - Let Iz be the identity function defined on the set...Ch. 7.1 - Find function defined on the sdet of nonnegative...Ch. 7.1 - Let A={1,2,3,4,5} , and define a function F:P(A)Z...Ch. 7.1 - Let Js={0,1,2,3,4} , and define a function F:JsJs...Ch. 7.1 - Define a function S:Z+Z+ as follows: For each...Ch. 7.1 - Prob. 10ES Ch. 7.1 - Define F:ZZZZ as follows: For every ordered pair...Ch. 7.1 - Let JS={0,1,2,3,4} ,and define G:JsJsJsJs as...Ch. 7.1 - Let Js={0,1,2,3,4} , and define functions f:JsJs...Ch. 7.1 - Define functions H and K from R to R by the...Ch. 7.1 - Prob. 15ES Ch. 7.1 - Let F and G be functions from the set of all real...Ch. 7.1 - Prob. 17ES Ch. 7.1 - Find exact values for each of the following...Ch. 7.1 - Prob. 19ES Ch. 7.1 - Prob. 20ES Ch. 7.1 - If b is any positive real number with b1 and x is...Ch. 7.1 - Prob. 22ES Ch. 7.1 - Prob. 23ES Ch. 7.1 - If b and y are positivereal numbers such that...Ch. 7.1 - Let A={2,3,5} and B={x,y}. Let p1 and p2 be the...Ch. 7.1 - Observe that mod and div can be defined as...Ch. 7.1 - Let S be the set of all strings of as and bs....Ch. 7.1 - Consider the coding and decoding functions E and D...Ch. 7.1 - Consider the Hamming distance function defined in...Ch. 7.1 - Draw arrow diagram for the Boolean functions...Ch. 7.1 - Fill in the following table to show the values of...Ch. 7.1 - Cosider the three-place Boolean function f defined...Ch. 7.1 - Student A tries to define a function g:QZ by the...Ch. 7.1 - Student C tries to define a function h:QQ by the...Ch. 7.1 - Let U={1,2,3,4} . Student A tries to define a...Ch. 7.1 - Prob. 36ES Ch. 7.1 - On certain computers the integer data type goed...Ch. 7.1 - Prob. 38ES Ch. 7.1 - Prob. 39ES Ch. 7.1 - Prob. 40ES Ch. 7.1 - Prob. 41ES Ch. 7.1 - In 41-49 let X and Y be sets, let A and B be any...Ch. 7.1 - Prob. 43ES Ch. 7.1 - Prob. 44ES Ch. 7.1 - Prob. 45ES Ch. 7.1 - Prob. 46ES Ch. 7.1 - Prob. 47ES Ch. 7.1 - Prob. 48ES Ch. 7.1 - Prob. 49ES Ch. 7.1 - Prob. 50ES Ch. 7.1 - Each of exercises 51-53 refers to the Euler phi...Ch. 7.1 - Prob. 52ES Ch. 7.1 - Each of exercises 51-53 refers to the Euler phi...Ch. 7.2 - If F is a function from a set X to a set Y, then F...Ch. 7.2 - If F is a function from a set X to a set Y, then F...Ch. 7.2 - Prob. 3TY Ch. 7.2 - Prob. 4TY Ch. 7.2 - Prob. 5TY Ch. 7.2 - Prob. 6TY Ch. 7.2 - Prob. 7TY Ch. 7.2 - Given a function F:XY , to prove that F is not one...Ch. 7.2 - Prob. 9TY Ch. 7.2 - Prob. 10TY Ch. 7.2 - Prob. 11TY Ch. 7.2 - The definition of onr-to-one is stated in two...Ch. 7.2 - Fill in each blank with the word most or least. a....Ch. 7.2 - When asked to state the definition of one-to-one,...Ch. 7.2 - Let f:XY be a function. True or false? A...Ch. 7.2 - All but two of the following statements are...Ch. 7.2 - Let X={1,5,9} and Y={3,4,7} . a. Define f:XY by...Ch. 7.2 - Let X={a,b,c,d} and Y={e,f,g} . Define functions F...Ch. 7.2 - Let X={a,b,c} and Y={d,e,f,g} . Define functions H...Ch. 7.2 - Let X={1,2,3},Y={1,2,3,4} , and Z= {1,2} Define a...Ch. 7.2 - a. Define f:ZZ by the rule f(n)=2n, for every...Ch. 7.2 - Define F:ZZZZ as follows. For every ordered pair...Ch. 7.2 - a. Define F:ZZ by the rule F(n)=23n for each...Ch. 7.2 - a. Define H:RR by the rule H(x)=x2 , for each real...Ch. 7.2 - Explain the mistake in the following “proof.”...Ch. 7.2 - In each of 15-18 a function f is defined on a set...Ch. 7.2 - Prob. 16ES Ch. 7.2 - Prob. 17ES Ch. 7.2 - Prob. 18ES Ch. 7.2 - Referring to Example 7.2.3, assume that records...Ch. 7.2 - Define Floor: RZ by the formula Floor (x)=x , for...Ch. 7.2 - Prob. 21ES Ch. 7.2 - Let S be the set of all strings of 0’s and 1’s,...Ch. 7.2 - Define F:P({a,b,c})Z as follaws: For every A in...Ch. 7.2 - Les S be the set of all strings of a’s and b’s,...Ch. 7.2 - Let S be the et of all strings is a’s and b’s, and...Ch. 7.2 - Prob. 26ES Ch. 7.2 - Let D be the set of all set of all finite subsets...Ch. 7.2 - Prob. 28ES Ch. 7.2 - Define H:RRRR as follows: H(x,y)=(x+1,2y) for...Ch. 7.2 - Define J=QQR by the rule J(r,s)=r+2s for each...Ch. 7.2 - Prob. 31ES Ch. 7.2 - a. Is log827=log23? Why or why not? b. Is...Ch. 7.2 - Prob. 33ES Ch. 7.2 - The properties of logarithm established in 33-35...Ch. 7.2 - Prob. 35ES Ch. 7.2 - Prob. 36ES Ch. 7.2 - Prob. 37ES Ch. 7.2 - Prob. 38ES Ch. 7.2 - Prob. 39ES Ch. 7.2 - Suppose F:XY is one—to—one. a. Prove that for...Ch. 7.2 - Suppose F:XY is into. Prove that for every subset...Ch. 7.2 - Prob. 42ES Ch. 7.2 - Prob. 43ES Ch. 7.2 - In 44-55 indicate which of the function in the...Ch. 7.2 - In 44-55 indicate which of the function in the...Ch. 7.2 - Prob. 46ES Ch. 7.2 - Prob. 47ES Ch. 7.2 - Prob. 48ES Ch. 7.2 - Prob. 49ES Ch. 7.2 - Prob. 50ES Ch. 7.2 - Prob. 51ES Ch. 7.2 - Prob. 52ES Ch. 7.2 - Prob. 53ES Ch. 7.2 - Prob. 54ES Ch. 7.2 - Prob. 55ES Ch. 7.2 - Prob. 56ES Ch. 7.2 - Write a computer algorithm to check whether a...Ch. 7.2 - Write a computer algorithm to check whether a...Ch. 7.3 - If f is a function from X to Y’,g is a function...Ch. 7.3 - Prob. 2TY Ch. 7.3 - If f is a one-to=-one correspondence from X to Y....Ch. 7.3 - Prob. 4TY Ch. 7.3 - Prob. 5TY Ch. 7.3 - Prob. 1ES Ch. 7.3 - In each of 1 and 2, functions f and g are defined...Ch. 7.3 - In 3 and 4, functions F and G are defined by...Ch. 7.3 - In 3 and 4, functions F and G are defined by...Ch. 7.3 - Define f:RR by the rule f(x)=x for every real...Ch. 7.3 - Define F:ZZ and G:ZZ . By the rules F(a)=7a and...Ch. 7.3 - Define L:ZZ and M:ZZ by the rules L(a)=a2 and...Ch. 7.3 - Let S be the set of all strings in a’s and b’s and...Ch. 7.3 - Define F:RR and G:RZ by the following formulas:...Ch. 7.3 - Prob. 10ES Ch. 7.3 - Define F:RR and G:RR by the rules F(n)=3x and...Ch. 7.3 - The functions of each pair in 12—14 are inverse to...Ch. 7.3 - G:R+R+ and G1:RR+ are defined by G(x)=x2andG1(x)=x...Ch. 7.3 - H and H-1 are both defined from R={1} to R-{1} by...Ch. 7.3 - Explain how it follows from the definition of...Ch. 7.3 - Prove Theorem 7.3.1(b): If f is any function from...Ch. 7.3 - Prove Theorem 7.3.2(b): If f:XY is a one-to-one...Ch. 7.3 - Prob. 18ES Ch. 7.3 - If + f:XY and g:YZ are functions and gf is...Ch. 7.3 - If f:XY and g:YZ are function and gf is onto, must...Ch. 7.3 - Prob. 21ES Ch. 7.3 - If f:XY and g:YZ are functions and gf is onto,...Ch. 7.3 - Prob. 23ES Ch. 7.3 - Prob. 24ES Ch. 7.3 - Prob. 25ES Ch. 7.3 - In 26 and 27 find (gf)1,g1,f1, and f1g1 , and...Ch. 7.3 - In 26 and 27 find (gf)1,g1,f1 , and f1g1 by the...Ch. 7.3 - Prob. 28ES Ch. 7.3 - Suppose f:XY and g:YZ are both one-to-one and...Ch. 7.3 - Prob. 30ES Ch. 7.4 - A set is finite if, and only if,________Ch. 7.4 - Prob. 2TY Ch. 7.4 - The reflexive property of cardinality says that...Ch. 7.4 - The symmetric property of cardinality says that...Ch. 7.4 - The transitive property of cardinality say that...Ch. 7.4 - Prob. 6TY Ch. 7.4 - Prob. 7TY Ch. 7.4 - Prob. 8TY Ch. 7.4 - Prob. 9TY Ch. 7.4 - Prob. 1ES Ch. 7.4 - Show that “there are as many squares as there are...Ch. 7.4 - Let 3Z={nZn=3k,forsomeintegerk} . Prove that Z and...Ch. 7.4 - Let O be the set of all odd integers. Prove that O...Ch. 7.4 - Let 25Z be the set of all integers that are...Ch. 7.4 - Prob. 6ES Ch. 7.4 - Prob. 7ES Ch. 7.4 - Use the result of exercise 3 to prove that 3Z is...Ch. 7.4 - Show that the set of all nonnegative integers is...Ch. 7.4 - In 10-14 s denotes the sets of real numbers...Ch. 7.4 - Prob. 11ES Ch. 7.4 - In 10-14 S denotes the set of real numbers...Ch. 7.4 - Prob. 13ES Ch. 7.4 - Prob. 14ES Ch. 7.4 - Show that the set of all bit string (string of 0’s...Ch. 7.4 - Prob. 16ES Ch. 7.4 - Prob. 17ES Ch. 7.4 - Must the average of two irrational numbers always...Ch. 7.4 - Prob. 19ES Ch. 7.4 - Give two examples of functions from Z to Z that...Ch. 7.4 - Give two examples of function from Z to Z that are...Ch. 7.4 - Define a function g:Z+Z+Z+ by the formula...Ch. 7.4 - âa. Explain how to use the following diagram to...Ch. 7.4 - Prob. 24ES Ch. 7.4 - Prob. 25ES Ch. 7.4 - Prove that any infinite set contain a countable...Ch. 7.4 - Prove that if A is any countably infinite set, B...Ch. 7.4 - Prove that a disjoint union of any finite set and...Ch. 7.4 - Prove that a union of any two countably infinite...Ch. 7.4 - Prob. 30ES Ch. 7.4 - Use the results of exercise 28 and 29 to prove...Ch. 7.4 - Prove that ZZ , the Cartesian product of the set...Ch. 7.4 - Prob. 33ES Ch. 7.4 - Let P(s) be the set of all subsets of set S, and...Ch. 7.4 - Prob. 35ES Ch. 7.4 - Prob. 36ES Ch. 7.4 - Prove that if A and B are any countably infinite...Ch. 7.4 - Prob. 38ES

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.