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Elements Of Modern Algebra

8th Edition
Gilbert + 2 others
ISBN: 9781285463230

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BuyFindarrow_forward

Elements Of Modern Algebra

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