A Transition to Advanced Mathematics
A Transition to Advanced Mathematics
8th Edition
ISBN: 9781285463261
Author: Douglas Smith, Maurice Eggen, Richard St. Andre
Publisher: Cengage Learning
bartleby

Concept explainers

bartleby

Videos

Question
Book Icon
Chapter 3.1, Problem 7E

a)

To determine

To find: the composite function RS .

a)

Expert Solution
Check Mark

Answer to Problem 7E

  RS={(3,5),(5,2)}

Explanation of Solution

Given Information:

Three sets are given as

  R={(1,5),(2,2),(3,4),(5,2)} , S={(2,4),(3,4),(3,1),(5,5)} and T={(1,4),(3,5),(4,1)}

Formula used:

Composition function of two sets R and S can be evaluated as

  RS(x)=R[S(x)]=R(y)

where x denote the domain of S and y denote the domain of R

Composition function of three sets R , S , and T can be evaluated as

  (RS)T(x)=RS[T(x)]=RS(y)=R[S(y)]=R(z)

where x is the domain of T and y is the domain of S and z is the domain of R

Calculation:

Consider the set R .

  R={(1,5),(2,2),(3,4),(5,2)}

Here in the pair (1,5) first element is in domain and second element is in range.

  Dom(R)={1,2,3,5} and Range(R)={2,4,5}

  R(1)=5,R(2)=2,R(3)=4 , and R(5)=2

Consider the set S .

  S={(2,4),(3,4),(3,1),(5,5)}

Here in the pair (2,4) first element is in domain and second element is in range.

  Dom(S)={2,3,5} and Range(S)={4,1,5}

  S(2)=4,S(3)=4,S(3)=1 , and S(5)=4

Construct the following diagram for the relation.

  A Transition to Advanced Mathematics, Chapter 3.1, Problem 7E , additional homework tip  1

Hence, RS={(3,5),(5,2)} .

b)

To determine

To find: the composite function RT .

b)

Expert Solution
Check Mark

Answer to Problem 7E

  RT={(3,2),(4,5)}

Explanation of Solution

Given Information:

Three sets are given as

  R={(1,5),(2,2),(3,4),(5,2)} , S={(2,4),(3,4),(3,1),(5,5)} and T={(1,4),(3,5),(4,1)}

Formula used:

Composition function of two sets R and S can be evaluated as

  RS(x)=R[S(x)]=R(y)

where x denote the domain of S and y denote the domain of R

Composition function of three sets R , S , and T can be evaluated as

  (RS)T(x)=RS[T(x)]=RS(y)=R[S(y)]=R(z)

where x is the domain of T and y is the domain of S and z is the domain of R

Calculation:

Consider the set R .

  R={(1,5),(2,2),(3,4),(5,2)}

Here in the pair (1,5) first element is in domain and second element is in range.

  Dom(R)={1,2,3,5} and Range(R)={2,4,5}

  R(1)=5,R(2)=2,R(3)=4 , and R(5)=2

Consider the set T .

  T={(1,4),(3,5),(4,1)}

  Dom(T)={1,3,4} and Range(T)={4,5,1}

  T(1)=4,T(3)=5 , and T(4)=1

Construct the following diagram for the relation.

  A Transition to Advanced Mathematics, Chapter 3.1, Problem 7E , additional homework tip  2

Hence, RT={(3,2),(4,5)} .

(c)

To determine

To find: the composite function TS .

(c)

Expert Solution
Check Mark

Answer to Problem 7E

  TS={(2,1),(3,4),(3,1)}

Explanation of Solution

Given Information:

Three sets are given as

  R={(1,5),(2,2),(3,4),(5,2)} , S={(2,4),(3,4),(3,1),(5,5)} and T={(1,4),(3,5),(4,1)}

Formula used:

Composition function of two sets R and S can be evaluated as

  RS(x)=R[S(x)]=R(y)

where x denote the domain of S and y denote the domain of R

Composition function of three sets R , S , and T can be evaluated as

  (RS)T(x)=RS[T(x)]=RS(y)=R[S(y)]=R(z)

where x is the domain of T and y is the domain of S and z is the domain of R

Calculation:

Consider the set S .

  S={(2,4),(3,4),(3,1),(5,5)}

Here in the pair (2,4) first element is in domain and second element is in range.

  Dom(S)={2,3,5} and Range(S)={4,1,5}

  S(2)=4,S(3)=4,S(3)=1 , and S(5)=4

Consider the set T .

  T={(1,4),(3,5),(4,1)}

  Dom(T)={1,3,4} and Range(T)={4,5,1}

  T(1)=4,T(3)=5 , and T(4)=1

Construct the following diagram for the relation.

  A Transition to Advanced Mathematics, Chapter 3.1, Problem 7E , additional homework tip  3

Hence, TS={(2,1),(3,4),(3,1)} .

(d)

To determine

To find: the composite function RR .

(d)

Expert Solution
Check Mark

Answer to Problem 7E

  RR={(1,2),(2,2),(5,2)}

Explanation of Solution

Given Information:

Three sets are given as

  R={(1,5),(2,2),(3,4),(5,2)} , S={(2,4),(3,4),(3,1),(5,5)} and T={(1,4),(3,5),(4,1)}

Formula used:

Composition function of two sets R and S can be evaluated as

  RS(x)=R[S(x)]=R(y)

where x denote the domain of S and y denote the domain of R

Composition function of three sets R , S , and T can be evaluated as

  (RS)T(x)=RS[T(x)]=RS(y)=R[S(y)]=R(z)

where x is the domain of T and y is the domain of S and z is the domain of R

Calculation:

Consider the set R .

  R={(1,5),(2,2),(3,4),(5,2)}

Here in the pair (1,5) first element is in domain and second element is in range.

  Dom(R)={1,2,3,5} and Range(R)={2,4,5}

  R(1)=5,R(2)=2,R(3)=4 , and R(5)=2

Construct the following diagram for the relation.

  A Transition to Advanced Mathematics, Chapter 3.1, Problem 7E , additional homework tip  4

Hence, RR={(1,2),(2,2),(5,2)} .

(e)

To determine

To find: the composite function SR .

(e)

Expert Solution
Check Mark

Answer to Problem 7E

  SR={(1,5),(2,4),(5,4)}

Explanation of Solution

Given Information:

Three sets are given as

  R={(1,5),(2,2),(3,4),(5,2)} , S={(2,4),(3,4),(3,1),(5,5)} and T={(1,4),(3,5),(4,1)}

Formula used:

Composition function of two sets R and S can be evaluated as

  RS(x)=R[S(x)]=R(y)

where x denote the domain of S and y denote the domain of R

Composition function of three sets R , S , and T can be evaluated as

  (RS)T(x)=RS[T(x)]=RS(y)=R[S(y)]=R(z)

where x is the domain of T and y is the domain of S and z is the domain of R

Calculation:

Consider the set R .

  R={(1,5),(2,2),(3,4),(5,2)}

Here in the pair (1,5) first element is in domain and second element is in range.

  Dom(R)={1,2,3,5} and Range(R)={2,4,5}

  R(1)=5,R(2)=2,R(3)=4 , and R(5)=2

Consider the set S .

  S={(2,4),(3,4),(3,1),(5,5)}

Here in the pair (2,4) first element is in domain and second element is in range.

  Dom(S)={2,3,5} and Range(S)={4,1,5}

  S(2)=4,S(3)=4,S(3)=1 , and S(5)=4

Construct the following diagram for the relation.

  A Transition to Advanced Mathematics, Chapter 3.1, Problem 7E , additional homework tip  5

Hence, SR={(1,5),(2,4),(5,4)} .

(f)

To determine

To find: the composite function TT .

(f)

Expert Solution
Check Mark

Answer to Problem 7E

  TT={(1,1),(4,4)}

Explanation of Solution

Given Information:

Three sets are given as

  R={(1,5),(2,2),(3,4),(5,2)} , S={(2,4),(3,4),(3,1),(5,5)} and T={(1,4),(3,5),(4,1)}

Formula used:

Composition function of two sets R and S can be evaluated as

  RS(x)=R[S(x)]=R(y)

where x denote the domain of S and y denote the domain of R

Composition function of three sets R , S , and T can be evaluated as

  (RS)T(x)=RS[T(x)]=RS(y)=R[S(y)]=R(z)

where x is the domain of T and y is the domain of S and z is the domain of R

Calculation:

Consider the set T .

  T={(1,4),(3,5),(4,1)}

  Dom(T)={1,3,4} and Range(T)={4,5,1}

  T(1)=4,T(3)=5 , and T(4)=1

Construct the following diagram for the relation.

  A Transition to Advanced Mathematics, Chapter 3.1, Problem 7E , additional homework tip  6

Hence, TT={(1,1),(4,4)} .

(g)

To determine

To find: the composite function R(ST) .

(g)

Expert Solution
Check Mark

Answer to Problem 7E

  R(ST)={(3,2)}

Explanation of Solution

Given Information:

Three sets are given as

  R={(1,5),(2,2),(3,4),(5,2)} , S={(2,4),(3,4),(3,1),(5,5)} and T={(1,4),(3,5),(4,1)}

Formula used:

Composition function of two sets R and S can be evaluated as

  RS(x)=R[S(x)]=R(y)

where x denote the domain of S and y denote the domain of R

Composition function of three sets R , S , and T can be evaluated as

  (RS)T(x)=RS[T(x)]=RS(y)=R[S(y)]=R(z)

where x is the domain of T and y is the domain of S and z is the domain of R

Calculation:

Consider the set R .

  R={(1,5),(2,2),(3,4),(5,2)}

Here in the pair (1,5) first element is in domain and second element is in range.

  Dom(R)={1,2,3,5} and Range(R)={2,4,5}

  R(1)=5,R(2)=2,R(3)=4 , and R(5)=2

Consider the set S .

  S={(2,4),(3,4),(3,1),(5,5)}

Here in the pair (2,4) first element is in domain and second element is in range.

  Dom(S)={2,3,5} and Range(S)={4,1,5}

  S(2)=4,S(3)=4,S(3)=1 , and S(5)=4

Consider the set T .

  T={(1,4),(3,5),(4,1)}

  Dom(T)={1,3,4} and Range(T)={4,5,1}

  T(1)=4,T(3)=5 , and T(4)=1

Construct the following diagram for the relation.

  A Transition to Advanced Mathematics, Chapter 3.1, Problem 7E , additional homework tip  7

  ST={(3,5)}

Now, find R(ST) as shown:

  (ST)(3)=5 , R(1)=5,R(2)=2,R(3)=4 , and R(5)=2

Construct the following diagram for the relation.

  A Transition to Advanced Mathematics, Chapter 3.1, Problem 7E , additional homework tip  8

Hence, R(ST)={(3,2)} .

(h)

To determine

To find: the composite function (RS)T .

(h)

Expert Solution
Check Mark

Answer to Problem 7E

  (RS)T={(3,2)}

Explanation of Solution

Given Information:

Three sets are given as

  R={(1,5),(2,2),(3,4),(5,2)} , S={(2,4),(3,4),(3,1),(5,5)} and T={(1,4),(3,5),(4,1)}

Formula used:

Composition function of two sets R and S can be evaluated as

  RS(x)=R[S(x)]=R(y)

where x denote the domain of S and y denote the domain of R

Composition function of three sets R , S , and T can be evaluated as

  (RS)T(x)=RS[T(x)]=RS(y)=R[S(y)]=R(z)

where x is the domain of T and y is the domain of S and z is the domain of R

Calculation:

Consider the set R .

  R={(1,5),(2,2),(3,4),(5,2)}

Here in the pair (1,5) first element is in domain and second element is in range.

  Dom(R)={1,2,3,5} and Range(R)={2,4,5}

  R(1)=5,R(2)=2,R(3)=4 , and R(5)=2

Consider the set S .

  S={(2,4),(3,4),(3,1),(5,5)}

Here in the pair (2,4) first element is in domain and second element is in range.

  Dom(S)={2,3,5} and Range(S)={4,1,5}

  S(2)=4,S(3)=4,S(3)=1 , and S(5)=4

Consider the set T .

  T={(1,4),(3,5),(4,1)}

  Dom(T)={1,3,4} and Range(T)={4,5,1}

  T(1)=4,T(3)=5 , and T(4)=1

Construct the following diagram for the relation.

  A Transition to Advanced Mathematics, Chapter 3.1, Problem 7E , additional homework tip  9

Hence, RS={(3,5),(5,2)} .

Now, find (RS)T as shown:

  (RS)(3)=5,(RS)(5)=2 , T(1)=4,T(3)=5 , and T(4)=1

Construct the following diagram for the relation.

  A Transition to Advanced Mathematics, Chapter 3.1, Problem 7E , additional homework tip  10

Hence, (RS)T={(3,2)} .

Want to see more full solutions like this?

Subscribe now to access step-by-step solutions to millions of textbook problems written by subject matter experts!
Students have asked these similar questions
A movie studio wishes to determine the relationship between the revenue generated from the streaming of comedies and the revenue generated from the theatrical release of such movies. The studio has the following bivariate data from a sample of fifteen comedies released over the past five years. These data give the revenue x from theatrical release (in millions of dollars) and the revenue y from streaming (in millions of dollars) for each of the fifteen movies. The data are displayed in the Figure 1 scatter plot. Theater revenue, x Streaming revenue, y (in millions of (in millions of dollars) dollars) 13.2 10.3 62.6 10.4 20.8 5.1 36.7 13.3 44.6 7.2 65.9 10.3 49.4 15.7 31.5 4.5 14.6 2.5 26.0 8.8 28.1 11.5 26.1 7.7 28.2 2.8 60.7 16.4 6.7 1.9 Streaming revenue (in millions of dollars) 18+ 16+ 14 12+ xx 10+ 8+ 6+ 2- 0 10 20 30 40 50 60 70 Theater revenue (in millions of dollars) Figure 1 Send data to calculator Send data to Excel The least-squares regression line for these data has a slope…
help on this, results given
a) Suppose that we are carrying out the 1-phase simplex algorithm on a linear program in standard inequality form (with 3 variables and 4 constraints) and suppose that we have reached a point where we have obtained the following tableau. Apply one more pivot operation, indicating the highlighted row and column and the row operations you carry out. What can you conclude from your updated tableau? x1 x2 x3 81 82 83 84 81 -2 0 1 1 0 0 0 3 82 3 0 -2 0 1 2 0 6 12 1 1 -3 0 0 1 0 2 84 -3 0 2 0 0 -1 1 4 -2 -2 0 11 0 0-4 0 -8

Chapter 3 Solutions

A Transition to Advanced Mathematics

Ch. 3.1 - Prove that if G is a group and H is a subgroup of...Ch. 3.1 - Prob. 12ECh. 3.1 - Prob. 13ECh. 3.1 - Prob. 14ECh. 3.1 - Prob. 15ECh. 3.1 - Prob. 16ECh. 3.1 - Prob. 17ECh. 3.2 - (a)Show that any two groups of order 2 are...Ch. 3.2 - (a)Show that the function h: defined by h(x)=3x is...Ch. 3.2 - Let R be the equivalence relation on ({0}) given...Ch. 3.2 - Let (R,+,) be an integral domain. Prove that 0 has...Ch. 3.2 - Complete the proof of Theorem 6.5.5. That is,...Ch. 3.2 - Prob. 6ECh. 3.2 - Assign a grade of A (correct), C (partially...Ch. 3.2 - Prob. 8ECh. 3.2 - Prob. 9ECh. 3.2 - Use the method of proof of Cayley's Theorem to...Ch. 3.2 - Prob. 11ECh. 3.2 - Assign a grade of A (correct), C (partially...Ch. 3.2 - Prob. 13ECh. 3.2 - Define on by setting (a,b)(c,d)=(acbd,ad+bc)....Ch. 3.2 - Prob. 15ECh. 3.2 - Let f:(A,)(B,*) and g:(B,*)(C,X) be OP maps. Prove...Ch. 3.2 - Prob. 17ECh. 3.2 - Let Conj: be the conjugate mapping for complex...Ch. 3.2 - Prove the remaining parts of Theorem 6.4.1.Ch. 3.3 - Let 3={3k:k}. Apply the Subring Test (Exercise...Ch. 3.3 - Use these exercises to check your understanding....Ch. 3.3 - Use these exercises to check your understanding....Ch. 3.3 - Use these exercises to check your understanding....Ch. 3.3 - Use these exercises to check your understanding....Ch. 3.3 - Prob. 6ECh. 3.3 - Use the definition of “divides” to explain (a) why...Ch. 3.3 - Prob. 8ECh. 3.3 - Prob. 9ECh. 3.3 - Complete the proof that for every m,(m+,) is a...Ch. 3.3 - Define addition and multiplication on the set ...Ch. 3.3 - Prob. 12ECh. 3.3 - Let (R,+,) be a ring and a,b,R. Prove that b+(a)...Ch. 3.3 - Prove the remaining parts of Theorem 6.5.3: For...Ch. 3.3 - We define a subring of a ring in the same way we...Ch. 3.4 - Prob. 1ECh. 3.4 - Prob. 2ECh. 3.4 - If possible, give an example of a set A such that...Ch. 3.4 - Let A. Prove that if sup(A) exists, then...Ch. 3.4 - Let A and B be subsets of . Prove that if sup(A)...Ch. 3.4 - a.Give an example of sets A and B of real numbers...Ch. 3.4 - a.Give an example of sets A and B of real numbers...Ch. 3.4 - An alternate version of the Archimedean Principle...Ch. 3.4 - Prob. 9ECh. 3.4 - Prob. 10ECh. 3.4 - Prob. 11ECh. 3.4 - Prob. 12ECh. 3.5 - Prob. 1ECh. 3.5 - Prob. 2ECh. 3.5 - Let A be a subset of . Prove that the set of all...Ch. 3.5 - Prob. 4ECh. 3.5 - Let be an associative operation on nonempty set A...Ch. 3.5 - Suppose that (A,*) is an algebraic system and * is...Ch. 3.5 - Let (A,o) be an algebra structure. An element lA...Ch. 3.5 - Let G be a group. Prove that if a2=e for all aG,...Ch. 3.5 - Give an example of an algebraic structure of order...Ch. 3.5 - Prove that an ordered field F is complete iff...Ch. 3.5 - Prove that every irrational number is "missing"...Ch. 3.5 - Find two upper bounds (if any exits) for each of...Ch. 3.5 - Prob. 13ECh. 3.5 - Prob. 14ECh. 3.5 - Prob. 15ECh. 3.5 - Let A and B be subsets of . Prove that if A is...Ch. 3.5 - Prob. 17ECh. 3.5 - Prob. 18ECh. 3.5 - Give an example of a set A for which both A and Ac...Ch. 3.5 - Prob. 20ECh. 3.5 - Prob. 21ECh. 3.5 - Prob. 22E
Knowledge Booster
Background pattern image
Advanced Math
Learn more about
Need a deep-dive on the concept behind this application? Look no further. Learn more about this topic, advanced-math and related others by exploring similar questions and additional content below.
Similar questions
SEE MORE QUESTIONS
Recommended textbooks for you
Text book image
Algebra & Trigonometry with Analytic Geometry
Algebra
ISBN:9781133382119
Author:Swokowski
Publisher:Cengage
Text book image
Elements Of Modern Algebra
Algebra
ISBN:9781285463230
Author:Gilbert, Linda, Jimmie
Publisher:Cengage Learning,
Text book image
Algebra: Structure And Method, Book 1
Algebra
ISBN:9780395977224
Author:Richard G. Brown, Mary P. Dolciani, Robert H. Sorgenfrey, William L. Cole
Publisher:McDougal Littell
Text book image
College Algebra
Algebra
ISBN:9781938168383
Author:Jay Abramson
Publisher:OpenStax
Text book image
Linear Algebra: A Modern Introduction
Algebra
ISBN:9781285463247
Author:David Poole
Publisher:Cengage Learning
What is a Function? Business Mathematics and Statistics; Author: Edmerls;https://www.youtube.com/watch?v=fcGNFyqRzuI;License: Standard YouTube License, CC-BY
FUNCTIONS CONCEPTS FOR CBSE/ISC/JEE/NDA/CET/BANKING/GRE/MBA/COMEDK; Author: Neha Agrawal Mathematically Inclined;https://www.youtube.com/watch?v=hhbYynJwBqk;License: Standard YouTube License, CC-BY