Explanation Formula used: A relation R on a set A is (i) Reflexive if ( a , a ) ∈ R for all a ∈ A (ii) Symmetric if ( a , b ) ∈ R implies that ( b , a ) ∈ R (iii) Transitive if ( a , b ) ∈ R and ( b , c ) ∈ R implies that ( a , c ) ∈ R (iv) An equivalence relation if it is reflexive, symmetric and transitive. Suppose nis a positive integer. Define a relation R on the set A (set of all polynomials with real-valued coefficient) as: R = { ( f , g ) | f ( n ) ( x ) = g ( n ) ( x ) } Now for any f ∈ A , we know that f ( n ) ( x ) = f ( n ) ( x ) for all x ∈ ℝ Thus, ( f , f ) ∈ R So, R is reflexive To determine (b) To find: The functions that are in the same equivalence class as the function f ( x ) = x 4 , where n = 3 .

8th Edition

ISBN: 9781260521337

Author: ROSEN

Publisher: MCG

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Chapter 9.5, Problem 18E

To determine

(a)

To show:

The relation R is an equivalence relation, whereR is a relationon the set of all polynomials with real-valued coefficient consisting of all pairs (f,g) such that f(n)(x)=g(n)(x).

To determine

(b)

To find:

The functions that are in the same equivalence class as the function f(x)=x4, where n=3.

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Chapter 9.5, Problem 17E

Chapter 9.5, Problem 19E

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Chapter 9 Solutions

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(a) To show: The relation R is an equivalence relation, whereR is a relationon the set of all polynomials with real-valued coefficient consisting of all pairs ( f , g ) such that f ( n ) ( x ) = g ( n ) ( x ) .

Solution Summary: The author explains that the relation R is an equivalence relation, whereR is a relationon the set of all polynomials with real-valued coefficient.

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(a)

To show:

The relation R is an equivalence relation, whereR is a relationon the set of all polynomials with real-valued coefficient consisting of all pairs (f,g) such that f(n)(x)=g(n)(x).

(b)

To find:

The functions that are in the same equivalence class as the function f(x)=x4, where n=3.

Want to see the full answer?

Chapter 9 Solutions

(a) To show: The relation R is an equivalence relation, whereR is a relationon the set of all polynomials with real-valued coefficient consisting of all pairs ( f , g ) such that f ( n ) ( x ) = g ( n ) ( x ) .

Solution Summary: The author explains that the relation R is an equivalence relation, whereR is a relationon the set of all polynomials with real-valued coefficient.

Videos

(a) To show: The relation R is an equivalence relation, whereR is a relationon the set of all polynomials with real-valued coefficient consisting of all pairs (f,g) such that f(n)(x)=g(n)(x).

(b) To find: The functions that are in the same equivalence class as the function f(x)=x4, where n=3.

Want to see the full answer?

Chapter 9 Solutions

(a)

To show:

The relation R is an equivalence relation, whereR is a relationon the set of all polynomials with real-valued coefficient consisting of all pairs (f,g) such that f(n)(x)=g(n)(x).

(b)

To find:

The functions that are in the same equivalence class as the function f(x)=x4, where n=3.