Chapter 1.4, Problem 1E

Elements Of Modern Algebra

8th Edition
Gilbert + 2 others
ISBN: 9781285463230

Chapter
Section

Elements Of Modern Algebra

8th Edition
Gilbert + 2 others
ISBN: 9781285463230
Textbook Problem
12 views

Decide whether the given set B is closed with respect to the binary operation defined on the set of integers Z . If B is not closed, exhibit elements x ∈ B and y ∈ B , such that x ∗ y ∉ B . a. x ∗ y = x y ,   B = { − 1 , − 2 , − 3 , … } b. x ∗ y = x − y ,   B = Z + c. x ∗ y = x 2 + y 2 ,   B = Z + d. x ∗ y = sgn x + sgn y ,   B = { − 2 , − 1 , 0 , 1 , 2 }   where   sgn x = {     1   if   x > 0     0   if   x = 0 − 1   if   x < 0 e. x ∗ y = | x | − | y | ,   B = Z + f. x ∗ y = x + x y ,   B = Z + g. x ∗ y = x y − x − y ,   B is the set of all odd integers. h. x ∗ y = x y ,   B is the set of positive odd integers.

a)

To determine

Whether the set B is closed with respect to the given binary operation defined on the sets of integers Z.

Explanation

Given Information:

The binary operation is defined as:

xy=xy,B={1,2,3,}

Explanation:

Consider the given set B={1,2,3,}

b)

To determine

Whether the set B is closed with respect to the given binary operation defined on the sets of integers Z.

c)

To determine

Whether the set B is closed with respect to the given binary operation defined on the sets of integers Z.

d)

To determine

Whether the set B is closed with respect to the given binary operation defined on the sets of integers Z.

e)

To determine

Whether the set B is closed with respect to the given binary operation defined on the sets of integers Z.

f)

To determine

Whether the set B is closed with respect to the given binary operation defined on the sets of integers Z.

g)

To determine

Whether the set B is closed with respect to the given binary operation defined on the sets of integers Z.

h)

To determine

Whether the set B is closed with respect to the given binary operation defined on the sets of integers Z.

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