6th Grade Math Textbook, Fundamentals

closure question

 &KDSWHU

1-8

Is the sum of two odd whole numbers

always odd?

Set:odd numbers

Operation:addition

Check: 1  5  6

6 is even.

Result:Since 6 is notodd, the set

of odd whole numbers is not

closed under addition.

Closure Property

Objective To identify the closure properties for any defined set of numbers

If performing an operation on anytwo numbers in a set alwaysresults

in a number that is in that set, then the set is closedunder that operation.

This is called the.

To test if a set of numbers is closed under an operation:

Identify the set of numbers and the operation

in the closure question.

Choose two numbers from the set.

Perform the given operation on those numbers

to check for closure.

Is the result in the set of numbers?

If yes, then that set could beclosed under that operation.

If no, then the set is notclosed under that operation.

Whenever the answer is no, the test case is called

a.

Explore the set of whole numbers for closure under

addition and subtraction.

counterexample

Closure Property

Key Concept

Counterexamples and

Confirming Examples

Only one counterexampleis needed

to prove that a set is notclosed

under a given operation. However,

one confirmingexample does not

prove that a set isclosed under a

given operation. If all possibilities

cannot be tested, then proof must

be established in some other way.

Is the set of whole numbers closed

under addition?

Set:whole numbers

Operation:addition

Check: 1  0  1

1 is a whole number.

Result:The set of whole numbers could

beclosed under addition.

(^1) Is the set of whole numbers closed
under subtraction?
Set:whole numbers
Operation:subtraction
Check: 6  10  4
4 is not a whole number.
Result:The set of whole numbers is not
closed under subtraction.
2
Mathematicians agree that the set of whole numbers is closed under
addition, but the set of whole numbers is not closed under subtraction.