Elements Of Modern Algebra
8th Edition
ISBN:9781285463230
Author:Gilbert, Linda, Jimmie
Publisher:Gilbert, Linda, Jimmie
Chapter1: Fundamentals
Section1.7: Relations
Problem 14E: In each of the following parts, a relation is defined on the set of all human beings. Determine...
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Question
Solve Q17 explaining detailly each step
![14. Define the concept of an equivalent relation. A relation R is defined on the set Z, of integers,
by:a R b a- a= b² – b. show that R is an equivalence relation. Write down 3 elements of
the relation R, where a R b but a + b.
15. A real valued function f is defined by f: x →
2х-3
X E R, x # -1
X+1
a) Obtain the composite function ff and state its domain
b) Show that f is injective
c) Find the range of f and deduce that f is not surjective
16. The functions f and g are defined by
x+1
f: x→*-, x € R, x #
3
|
2x+3
2
g: x→3x +2,x E R.
a) Express, in a similar manner, the function (gf)-1
b) Obtain an element in the domain of g which is invariant under g.
17. i) The functions f and g are defined on R, the set of real numbers, by
f: x H – , x # 1 and g: x 2x – 3.
X-1
Find the composite function f og, stating its domain.
ii) A binary relation R is defined on the set of integers, by: xRy x - y = 3c, wherec is
an integer. Prove that R is an equivalence relation.
18. A function f: R → R is defined by f: x →
3x
1
Determine whether or not f is
2x+1
2
surjective.
19. The function f: R R is defined by (x) = 1
x3
. Find the monotony of f, showing
|
3
clearly its variation table](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F5a6d9c67-6f13-49d2-ac4d-2d996f90a88b%2F7d1c3bcd-0492-4c91-9aa3-3c2a26f678c2%2Frygaao_processed.jpeg&w=3840&q=75)
Transcribed Image Text:14. Define the concept of an equivalent relation. A relation R is defined on the set Z, of integers,
by:a R b a- a= b² – b. show that R is an equivalence relation. Write down 3 elements of
the relation R, where a R b but a + b.
15. A real valued function f is defined by f: x →
2х-3
X E R, x # -1
X+1
a) Obtain the composite function ff and state its domain
b) Show that f is injective
c) Find the range of f and deduce that f is not surjective
16. The functions f and g are defined by
x+1
f: x→*-, x € R, x #
3
|
2x+3
2
g: x→3x +2,x E R.
a) Express, in a similar manner, the function (gf)-1
b) Obtain an element in the domain of g which is invariant under g.
17. i) The functions f and g are defined on R, the set of real numbers, by
f: x H – , x # 1 and g: x 2x – 3.
X-1
Find the composite function f og, stating its domain.
ii) A binary relation R is defined on the set of integers, by: xRy x - y = 3c, wherec is
an integer. Prove that R is an equivalence relation.
18. A function f: R → R is defined by f: x →
3x
1
Determine whether or not f is
2x+1
2
surjective.
19. The function f: R R is defined by (x) = 1
x3
. Find the monotony of f, showing
|
3
clearly its variation table
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