# Closure property is satisfied in whole numbers with respect to which of the following?

A.Addition and subtraction

B.Addition and division

C.Addition and multiplication

D.Multiplication and division

Answer

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Hint: First define the closure property and then check for the given operations for satisfying the closure property in whole numbers.

Complete step-by-step answer:

A set is closed under an operation if performance of that operation on members of the set always produces a member of that set only.

Given in the problem, we are given whole numbers.

Whole numbers are the set of positive integers that include zero.

Closure property of an operation in whole numbers means that if $x$ and $y$ are two whole numbers, then the operation $*$ satisfies the closure property if the result of $x*y$ is also a whole number.

We need to test the closure property of whole numbers with respect to addition, multiplication, subtraction and division.

Performing the operations one by one.

In case of addition , if we add two whole numbers say $a$ and $b$ such that $a + b = c$, their sum $c$ is always a whole number.

For example: $3 + 4 = 7,0 + 0 = 0$,etc.

Hence closure property for addition in whole numbers is always true.

In case of subtraction , if we subtract two whole numbers say $a$ and $b$ such that $a - b = c$, their difference $c$ is need not to be always a whole number.

For example: $3 - 4 = - 1$ , which is not a whole number.

Hence closure property for subtraction in whole numbers is not always satisfied.

In case of division , if we divide two whole numbers say $a$ and $b$ such that $a \div b = c$, their quotient $c$ is need not to be always a whole number.

For example: $3 \div 4 = 0.75$ , which is not a whole number.

Hence closure property for division in whole numbers is not always satisfied.

Lastly in case of multiplication , if we multiply two whole numbers say $a$ and $b$ such that $a \times b = c$, their product $c$ is always a whole number.

For example: $3 \times 4 = 12,0 \times 0 = 0$,etc.

Hence closure property for multiplication in whole numbers is always true.

Hence closure property is satisfied in whole numbers with respect to addition and multiplication.

Therefore, option (C). Addition and multiplication are the correct answer.

Note: The definition of whole numbers and closure property should be kept in mind in problems like above. One should try to look up examples in order to contradict an operation for given conditions in problems like above.

Complete step-by-step answer:

A set is closed under an operation if performance of that operation on members of the set always produces a member of that set only.

Given in the problem, we are given whole numbers.

Whole numbers are the set of positive integers that include zero.

Closure property of an operation in whole numbers means that if $x$ and $y$ are two whole numbers, then the operation $*$ satisfies the closure property if the result of $x*y$ is also a whole number.

We need to test the closure property of whole numbers with respect to addition, multiplication, subtraction and division.

Performing the operations one by one.

In case of addition , if we add two whole numbers say $a$ and $b$ such that $a + b = c$, their sum $c$ is always a whole number.

For example: $3 + 4 = 7,0 + 0 = 0$,etc.

Hence closure property for addition in whole numbers is always true.

In case of subtraction , if we subtract two whole numbers say $a$ and $b$ such that $a - b = c$, their difference $c$ is need not to be always a whole number.

For example: $3 - 4 = - 1$ , which is not a whole number.

Hence closure property for subtraction in whole numbers is not always satisfied.

In case of division , if we divide two whole numbers say $a$ and $b$ such that $a \div b = c$, their quotient $c$ is need not to be always a whole number.

For example: $3 \div 4 = 0.75$ , which is not a whole number.

Hence closure property for division in whole numbers is not always satisfied.

Lastly in case of multiplication , if we multiply two whole numbers say $a$ and $b$ such that $a \times b = c$, their product $c$ is always a whole number.

For example: $3 \times 4 = 12,0 \times 0 = 0$,etc.

Hence closure property for multiplication in whole numbers is always true.

Hence closure property is satisfied in whole numbers with respect to addition and multiplication.

Therefore, option (C). Addition and multiplication are the correct answer.

Note: The definition of whole numbers and closure property should be kept in mind in problems like above. One should try to look up examples in order to contradict an operation for given conditions in problems like above.

Last updated date: 27th Sep 2023

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