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Which of the following binary operations are closed?
a. subtraction of positive integers
b. division of nonzero integers
c. function composition of polynomials with real coefficients
d. multiplication of
e. exponentiation of integers
a.
Whether the given set is closed or not.
Answer to Problem 1E
The given set of positive integers is not closed under subtraction operation.
Explanation of Solution
Given:
Set of positive integers and operation is subtraction.
Concept Used:
An set G is said to be closed under operation
Calculation:
Given setof positive integers is not closedunder the operation subtraction as because there exist some elements
The result,
Hence the given set of positive integers is not closed under subtraction operation.
Conclusion:
The given setof positive integers is not closed under subtraction operation.
b.
Whether the given set is closed or not.
Answer to Problem 1E
The set of non zero integers is not closed under division operation.
Explanation of Solution
Given:
Set is non zero integers and operation is division
Concept Used:
An set G is said to be closed under operation
Calculation:
Given set non zero of integers is not closed under the operation division as because there exist some elements
The result
Hence the given set of non zero integers is not closed under division operation.
Conclusion:
The the given set of non zero integers is not closed under division operation
c.
Whether the given set is closed or not.
Answer to Problem 1E
The given set of polynomial functions with real coefficients is closed under composion.
Explanation of Solution
Given:
Set of polynomial functions with real coefficients and operation is composition.
Concept Used:
An set G is said to be closed under operation
Calculation:
Given set is closed under the operation composition
Thus
Hence the Set of polynomial functions with real coefficients is closed under the operation composition.
Conclusion:
The given the Set of polynomial functions with real coefficients is closed under the operation composition.
d.
Whether the given set is closed or not.
Answer to Problem 1E
The given set of matrix of order
Explanation of Solution
Given:
Set of matrix of order
Concept Used:
An set G is said to be closed under operation
Calculation:
Given set of matrix of order
Thus
Hence the given set of matrix of order
Conclusion:
The given set of matrix of order
e.
Whether the given set is closed or not.
Answer to Problem 1E
The given set of integers is not closed under exponent operation.
Explanation of Solution
Given:
Set of integers and operation is exponent.
Concept Used:
An set G is said to be closed under operation
Calculation:
Given set of integers is not closed under the exponent as because there exist some elements
The result,
Hence the given set of integers is not closed under exponent operation.
Conclusion:
The given set of integers is not closed under exponent operation.
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Chapter 2 Solutions
Contemporary Abstract Algebra
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