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
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Chapter 3.5, Problem 1E

(a)

To determine

To find whether therelation R is antisymmetric and has comparability property.

(a)

Expert Solution
Check Mark

Answer to Problem 1E

The given relation Ris neither antisymmetric nor has comparability property.

Explanation of Solution

Given information: The relation R defined on the set A={1,2,3,4,5} is R={(1,3),(1,1),(2,4),(3,2),(5,4),(4,2),}

  A={1,2,3,4,5} .

Definition used:

  • A relation R on a set A is said to have comparability property then, every two elements on Ashould be comparable.
  • A relation R on a set Ais antisymmetric if and only if for all x,yA,if xRy and xRy, thenx=y

Calculation:

The given relation is not antisymmetric since, (2,4) and (4,2) belongs to R but 24

The given relation R doesn’t have comparability property since (1,2),(2,5),(1,5),(5,3) doesn’t belong to R.

b.

To determine

To find whether the relation R is antisymmetric and has comparability property.

b.

Expert Solution
Check Mark

Answer to Problem 1E

The given relation R is antisymmetric and doesn’t have comparability property

Explanation of Solution

Given information: The relation R defined on the set A={1,2,3,4,5} is {(1,4),(1,2),(2,3),(3,4),(5,2),(4,2),(1,3)} .

Definition used:

  • A relation R on a set A is said to have comparability property then, every two elements on Ashould be comparable.
  • A relation R on a set A is antisymmetric if and only if for all x,yA,if xRy and xRy, thenx=y

Calculation:

The given relation is antisymmetric since, for every x,yA,xRy but there doesnot

exist yRx .

The given relation doesn’t have comparability property since (1,5),(3,5),(4,5) doesn’t belong to the relation.

c.

To determine

To find whether the relation R is antisymmetric and has comparability property.

c.

Expert Solution
Check Mark

Answer to Problem 1E

The given relation R is not antisymmetric and doesn’t have comparability property

Explanation of Solution

Given information: The relation R defined on the set is xRy if x2=y2 .

Definition used:

  • A relation R on a set A is said to have comparability property then, every two elements on Ashould be comparable.
  • A relation R on a set A is antisymmetric if and only if for all x,yA,if xRy and xRy, thenx=y

Calculation:

The given relation is not antisymmetric.

Example: Let x=3 and y=3

  (3)2=9=(x)2,xRyand (3)2=9=(3)2,yRx

But xy

The given relation doesn’t have comparability property.

Example: 1,2A  but 1222 . Hence (1,2)R .

d.

To determine

To find whether the relation R is antisymmetric and has comparability property.

d.

Expert Solution
Check Mark

Answer to Problem 1E

The given relation R is not antisymmetric and has comparability property

Explanation of Solution

Given information: The relation R defined on the set   is xRy if  x2y .

Definition used:

  • A relation R on a set A is said to have comparability property then, every two elements on A should be comparable.
  • A relation R on a set A is antisymmetric if and only if for all x,yA,if xRy and xRy, thenx=y

Calculation:

The given relation is not antisymmetric.

Example: Let x=3 and y=4

  324, so xRy423, so yRxBut 3

The given relation has comparability property because for all x,y either xRyor yRx .

e.

To determine

To find whether the relation R is antisymmetric and has comparability property.

e.

Expert Solution
Check Mark

Answer to Problem 1E

The given relation R is antisymmetric and doesn’t have comparability property.

Explanation of Solution

Given information: The relation R defined on the set × is xSyif y=x1 .

Definition used:

  • A relation R on a set A is said to have comparability property then, every two elements on A should be comparable.
  • A relation R on a set A is antisymmetric if and only if for all x,yA,if xRy and xRy, thenx=y

Calculation:

The given relation is antisymmetric.

For x,y× , y=x1and x=y1 cannot happen simultaneously.

In other words, if y=x1and x=y1 , then x=y is true for all x,y× .

The given relation doesn’t have comparability property since there are indefinite numbers in × such that y=x1and x=y1 is false.

f.

To determine

To find whether the relation R is antisymmetric and has comparability property.

f.

Expert Solution
Check Mark

Answer to Problem 1E

The given relation R is not antisymmetric and doesn’t have comparability property

Explanation of Solution

Given information: The relation R defined on the set A={1,2,3,4} is shown in the digraph below:

  A Transition to Advanced Mathematics, Chapter 3.5, Problem 1E , additional homework tip  1

Definition used:

  • A relation R on a set A is said to have comparability property then, every two elements on A should be comparable.
  • A relation R on a set A is antisymmetric if and only if for all x,yA,if xRy and xRy, thenx=y

Calculation:

From the digraph, the relation set can be obtained as:

  R={(1,1),(1,4),(4,4),(4,3),(3,3),(3,2),(2,2),(2,1),(1,3),(3,1)}

The given relation is not antisymmetric.

Example:

  (1,3),(3,1)R , but 13 .

The given relation doesn’t have comparability:

For example (2,4)or (4,2) doesn’t belong to the relation.

g.

To determine

To find whether the relation R is antisymmetric and has comparability property.

g.

Expert Solution
Check Mark

Answer to Problem 1E

The given relation R is antisymmetric and has comparability property

Explanation of Solution

The given relation is not antisymmetric sinceGiven information: The relation R defined on the set A={1,2,3,4} is shown in digraph as:

  A Transition to Advanced Mathematics, Chapter 3.5, Problem 1E , additional homework tip  2

Definition used:

A relation R on a set A is said to have comparability property then, every two elements on A should be comparable.

A relation R on a set A is antisymmetric if and only if for all x,yA,if xRy and xRy, thenx=y

Calculation:

From the digraph, the relation set can be obtained as:

  R={(1,2),(2,3),(3,4),(4,1),(2,4),(1,3)}

The given relation is antisymmetric since for every x,yAandxRy but there doesn’t exist yRx

The given relation have comparability property since for every x,yA,either

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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
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