Chapter 2, Problem 1E

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 $2\times 2$ matrices with integer entries
e. exponentiation of integers

To determine

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 $\ast$ , If $a\ast b\in G,\forall a,b\in G$ -------(1)

Calculation:

Given setof positive integers is not closedunder the operation subtraction as because there exist some elements $a=3,b=5$ in $G$ of positive integers,

The result, $a-b=3-5=-2$ is not belongs topositive integers

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.

To determine

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 $\ast$ , If $a\ast b\in G,\forall a,b\in G$ -------(1)

Calculation:

Given set non zero of integers is not closed under the operation division as because there exist some elements $a=8,b=12$ in $G$ of non zero integers,

The result $\frac{8}{12}=\frac{2}{3}$ is not belongs to in non zero integers.

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

To determine

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 $\ast$ , If $a\ast b\in G,\forall a,b\in G$ -------(1)

Calculation:

Given set is closed under the operation composition $\circ$ as because for every elements of $a=f(x),b=g(x)$ in $G$ of polynomial functions is again in G of polynomial functions.

Thus $f(x)\circ g(x)\in G,\forall f(x),g(x)\in G$

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.

To determine

Whether the given set is closed or not.

Answer to Problem 1E

The given set of matrix of order $2\times 2$ with integer’s entries is closed under multiplication operation.

Explanation of Solution

Given:

Set of matrix of order $2\times 2$ with integer’s entries and operation is multiplication.

Concept Used:

An set G is said to be closed under operation $\ast$ , If $A\ast B\in G,\forall A,B\in G$ -------(1)

Calculation:

Given set of matrix of order $2\times 2$ with integer’s entries is closed as because for every elements of $A,B$ in $G$ , the result $A\times B$ is again in G.

Thus $A\times B\in G,\forall A,B\in G$

Hence the given set of matrix of order $2\times 2$ with integer’s entries is closed underoperation multiplication.

Conclusion:

The given set of matrix of order $2\times 2$ with integer’s entries is closed under multiplication operation.

To determine

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 $\ast$ , If $a^n\in G,\forall a,n\in G$ -------(1)

Calculation:

Given set of integers is not closed under the exponent as because there exist some elements $a=2,b=-2$ in $G$ of integers,

The result, $a^n=2^{-2}=\frac{1}{2^2}=\frac{1}{4}$ is not belongs integers.

Hence the given set of integers is not closed under exponent operation.

Conclusion:

The given set of integers is not closed under exponent operation.