Question

use these four expressions to make a hypothesis about the closure property of integers under each operation.
select yes or no for each operation.

operation	example expression	is the set of integers closed under the operation?
subtraction	9 - (-11)	yes no
multiplication	3 · (-4)	yes no
division	-1 ÷ 6	yes no

Explanation:

To determine if the set of integers is closed under each operation, we analyze each one:

Addition:

The example is \(-8 + 5\). Calculating: \(-8 + 5 = -3\), which is an integer. Integers are closed under addition (the sum of two integers is always an integer). So the answer is yes.

Subtraction:

The example is \(9 - (-11)\). Simplifying: \(9 - (-11) = 9 + 11 = 20\), which is an integer. Integers are closed under subtraction (the difference of two integers is always an integer). So the answer is yes.

Multiplication:

The example is \(3 \cdot (-4)\). Calculating: \(3 \cdot (-4) = -12\), which is an integer. Integers are closed under multiplication (the product of two integers is always an integer). So the answer is yes.

Division:

The example is \(-1 \div 6\). Calculating: \(-1 \div 6 = -\frac{1}{6}\), which is not an integer (it is a fraction). Integers are not closed under division (the quotient of two integers is not always an integer). So the answer is no.

Final Answers:

• Addition: yes
• Subtraction: yes
• Multiplication: yes
• Division: no

(In the table, select "yes" for addition, subtraction, multiplication; select "no" for division.)

Answer:

To determine if the set of integers is closed under each operation, we analyze each one:

Addition:

The example is \(-8 + 5\). Calculating: \(-8 + 5 = -3\), which is an integer. Integers are closed under addition (the sum of two integers is always an integer). So the answer is yes.

Subtraction:

The example is \(9 - (-11)\). Simplifying: \(9 - (-11) = 9 + 11 = 20\), which is an integer. Integers are closed under subtraction (the difference of two integers is always an integer). So the answer is yes.

Multiplication:

The example is \(3 \cdot (-4)\). Calculating: \(3 \cdot (-4) = -12\), which is an integer. Integers are closed under multiplication (the product of two integers is always an integer). So the answer is yes.

Division:

The example is \(-1 \div 6\). Calculating: \(-1 \div 6 = -\frac{1}{6}\), which is not an integer (it is a fraction). Integers are not closed under division (the quotient of two integers is not always an integer). So the answer is no.

Final Answers:

• Addition: yes
• Subtraction: yes
• Multiplication: yes
• Division: no

(In the table, select "yes" for addition, subtraction, multiplication; select "no" for division.)