
A Transition to Advanced Mathematics
8th Edition
ISBN: 9781285463261
Author: Douglas Smith, Maurice Eggen, Richard St. Andre
Publisher: Cengage Learning
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Question
Chapter 2.6, Problem 12E
a.
To determine
Prove the theorem
b.
To determine
Prove the theorem
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You manage a chemical company with 2 warehouses. The following quantities of
Important Chemical A have arrived from an international supplier at 3 different
ports:
Chemical Available (L)
Port 1
Port 2
Port 3
400
110
100
The following amounts of Important Chemical A are required at your warehouses:
Warehouse 1
Warehouse 2
Chemical Required (L)
380
230
The cost in £ to ship 1L of chemical from each port to each warehouse is as follows:
Warehouse 1 Warehouse 2
Port 1
£10
£45
Port 2
£20
£28
Port 3
£13
£11
(a) You want to know how to send these shipments as cheaply as possible. For-
mulate this as a linear program (you do not need to formulate it in standard
inequality form) indicating what each variable represents.
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 12 23
81
82
83
S4
$1
-20
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
b) Solve the following linear program using the 2-phase simplex algorithm. You should give
the initial tableau and each further tableau produced during the execution of the
algorithm. If the program has an optimal solution, give this solution and state its
objective value. If it does not have an optimal solution, say why.
maximize 21 - - 2x2 + x3 - 4x4
subject to 2x1+x22x3x4≥ 1,
5x1+x2-x3-4 -1,
2x1+x2-x3-342,
1, 2, 3, 4 ≥0.
Suppose we have a linear program in standard equation form
maximize c'x
subject to Ax=b,
x≥ 0.
and suppose u, v, and w are all optimal solutions to this linear program.
(a) Prove that zu+v+w is an optimal solution.
(b) If you try to adapt your proof from part (a) to prove that that u+v+w
is an optimal solution, say exactly which part(s) of the proof go wrong.
(c) If you try to adapt your proof from part (a) to prove that u+v-w is an
optimal solution, say exactly which part(s) of the proof go wrong.
Chapter 2 Solutions
A Transition to Advanced Mathematics
Ch. 2.1 - The Cayley tables for operations o,*,+, and are...Ch. 2.1 - Let m,n and M=A:A is an mn matrix with real number...Ch. 2.1 - Let be an associative operation on nonempty set A...Ch. 2.1 - Let be an associative operation on nonempty set A...Ch. 2.1 - Suppose that (A,*) is an algebraic system and * is...Ch. 2.1 - Let (A,o) be an algebra structure. An element lA...Ch. 2.1 - Let G be a group. Prove that if a2=e for all aG,...Ch. 2.1 - Give an example of an algebraic structure of order...Ch. 2.1 - Prob. 9ECh. 2.1 - Construct the operation table for each of the...
Ch. 2.1 - Prob. 11ECh. 2.1 - Prob. 12ECh. 2.1 - Suppose m and m2. Prove that 1 and m1 are distinct...Ch. 2.1 - Let m and a be natural numbers with am. Complete...Ch. 2.1 - Complete the proof of Theorem 6.1.4. First, show...Ch. 2.1 - Prob. 16ECh. 2.1 - Prob. 17ECh. 2.1 - Prob. 18ECh. 2.1 - Repeat Exercise 2 with the operation * given by...Ch. 2.2 - Prob. 1ECh. 2.2 - Let G be a group and aiG for all n. Prove that...Ch. 2.2 - Prove part (d) of Theorem 6.2.3. That is, prove...Ch. 2.2 - Prove part (b) of Theorem 6.2.4.Ch. 2.2 - List all generators of each cyclic group in...Ch. 2.2 - Let G be a group with identity e. Let aG. Prove...Ch. 2.2 - Let G be a group, and let H be a subgroup of G....Ch. 2.2 - Let ({0},) be the group of nonzero complex numbers...Ch. 2.2 - Prove that for every natural number m greater than...Ch. 2.2 - Show that the structure ({1},), with operation ...Ch. 2.2 - (a)In the group G of Exercise 2, find x such that...Ch. 2.2 - Show that (,), with operation # defined by...Ch. 2.2 - Prob. 13ECh. 2.2 - Prob. 14ECh. 2.2 - Prob. 15ECh. 2.2 - Show that each of the following algebraic...Ch. 2.2 - Prob. 17ECh. 2.2 - Given that G={e,u,v,w} is a group of order 4 with...Ch. 2.2 - Give an example of an algebraic system (G,o) that...Ch. 2.2 - (a)What is the order of S4, the symmetric group on...Ch. 2.3 - Find the order of the element 3 in each group....Ch. 2.3 - Find the order of each element of the group S3....Ch. 2.3 - Let 3 and 6 be the sets of integer multiples of 3...Ch. 2.3 - Let (3,+) and (6,+) be the groups in Exercise 10,...Ch. 2.3 - Let ({a,b,c},o) be the group with the operation...Ch. 2.3 - (a)Prove that the function f:1824 given by f(x)=4x...Ch. 2.3 - Define f:1512 by f(x)=4x. Prove that f is a...Ch. 2.3 - Let (G,) and (H,*) be groups, i be the identity...Ch. 2.3 - Show that (4,+) and ({1,1,i,i},) are isomorphic.Ch. 2.3 - Prove that every subgroup of a cyclic group is...Ch. 2.3 - Let G=a be a cyclic group of order 30. What is the...Ch. 2.3 - Assign a grade of A (correct), C (partially...Ch. 2.3 - Find all subgroups of (8,+). (U11,). (5,+). (U7,)....Ch. 2.3 - In the group S4, find two different subgroups that...Ch. 2.3 - Prove that if G is a group and H is a subgroup of...Ch. 2.3 - (a)Prove that if H and K are subgroups of a group...Ch. 2.3 - Let G be a group and H be a subgroup of G. If H is...Ch. 2.3 - Prove or disprove: Every abelian group is cyclic.Ch. 2.3 - Let G be a group. If H is a subgroup of G and K is...Ch. 2.4 - Define f:++ by f(x)=x where + is the set of all...Ch. 2.4 - Assign a grade of A (correct), C (partially...Ch. 2.4 - Define f: by f(x)=x3. Is f:(,+)(,+) operation...Ch. 2.4 - Define on by setting (a,b)(c,d)=(acbd,ad+bc)....Ch. 2.4 - Let f the set of all real-valued integrable...Ch. 2.4 - Prob. 6ECh. 2.4 - Let M be the set of all 22 matrices with real...Ch. 2.4 - Let Conj: be the conjugate mapping for complex...Ch. 2.4 - Prove the remaining parts of Theorem 6.4.1.Ch. 2.4 - Is S3 isomorphic to (6,+)? Explain.Ch. 2.4 - Prob. 11ECh. 2.4 - Use the method of proof of Cayley's Theorem to...Ch. 2.5 - Let (R,+,) be an algebraic structure such that...Ch. 2.5 - Assign a grade of A (correct), C (partially...Ch. 2.5 - Which of the following is a ring with the usual...Ch. 2.5 - Let [2] be the set {a+b2:a,b}. Define addition and...Ch. 2.5 - Complete the proof that for every m,(m+,) is a...Ch. 2.5 - Define addition and multiplication on the set ...Ch. 2.5 - Prob. 7ECh. 2.5 - Let (R,+,) be a ring and a,b,R. Prove that b+(a)...Ch. 2.5 - Prove the remaining parts of Theorem 6.5.3: For...Ch. 2.5 - Prob. 10ECh. 2.5 - Prob. 11ECh. 2.5 - Prob. 12ECh. 2.5 - Prob. 13ECh. 2.5 - Prob. 14ECh. 2.6 - Prob. 1ECh. 2.6 - Let A and B be subsets of . Prove that if sup(A)...Ch. 2.6 - (a)Give an example of sets A and B of real numbers...Ch. 2.6 - (a)Give an example of sets A and B of real numbers...Ch. 2.6 - Prob. 5ECh. 2.6 - Prob. 6ECh. 2.6 - Prob. 7ECh. 2.6 - Prob. 8ECh. 2.6 - Prob. 9ECh. 2.6 - Prob. 10ECh. 2.6 - Prob. 11ECh. 2.6 - Prob. 12ECh. 2.6 - Prob. 13ECh. 2.6 - Prob. 14ECh. 2.6 - Prob. 15ECh. 2.6 - Prob. 16ECh. 2.6 - Use the definition of “divides” to explain (a) why...Ch. 2.6 - Prob. 18ECh. 2.6 - Prob. 19ECh. 2.6 - Prob. 20ECh. 2.6 - For each function, find the value of f at 3 and...Ch. 2.6 - Let A be the set {1,2,3,4} and B={0,1,2,3}. Give a...Ch. 2.6 - Formulate and prove a characterization of greatest...Ch. 2.6 - Prob. 24ECh. 2.6 - Prob. 25E
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