Question

Under what operations are the set of integers closed? Explain your answer. UPDATE ANSWER IS

-The set of integers is closed under addition, subtraction, and multiplication.
-The set of integers is not closed under division.
-Give a counterexample to show that the integers are not closed under division. For example, 1 divided by 3 is not an integer.

Answer 1

see below

Explanation:

Closed under addition, subtraction and multiplication

Not closed under division :-

for example 3/4 = 0.75 which is not an integer

Answer 2

Closure Property(Integers)

1. Addition : Adding two integers gives an Integer, so Integer satisfy closure property.For example

2+3=5

2 and 3 are integers,so 2+3=5 , will be also an integer.

2.Subtraction : Subtracting an integer from another one yields an integer.So, Integer satisfy closure property with respect to Subtraction.

For example

2-3=-1

2 and 3 are integers,so 2-3= -1 , will be also an integer.

3.Multiplication: Multiplication of an Intreger with another integer result in an integer.So, Integer satisfy closure property with respect to Multiplication.

For example

2×3=6

2 and 3 are integers,so 2×3=6 , will be also an integer.

4.

Integers are not closed under divison.Because

`\rightarrow (a)/(b)\\eq (b)/(a)\\\\(2)/(3)\\eq (3)/(2)`