Algebra/Closure

Closure

Closure is a property that is defined for a set of numbers and an operation. This Wikipedia article gives a description of the closure property with examples from various areas in math. As an Algebra student being aware of the closure property can help you solve a problem. For instance a problem might state "The sum of two whole numbers is 24." With practice you will be able to see that the possible set of numbers will be either all odd (e.g. (1,23),(3,21), ... etc.) or all even (e.g. (2,22), (4,20), ... etc.). The problem might not explicitly state the idea of whole numbers. It might state that two sides of a square sum to 24. If you remember working a problem like this before you know that the sides of a square need to be equal and you divide by two. The author of the problem might want to be trickier and say that two sides of an equilateral triangle sum to 24 and then ask for the perimeter of the triangle. In this case you might want to write the equation $3x=p$ for the perimeter of an equilateral triangle. This might make it easier for you to see that again you just need to divide 24 by 2 to find the length of one side and plug it into the equation.

Exercises For Closure

Consider each statement and try to come up with an example that proves it is false. If you can do that write down your example and mark the answer false. Otherwise mark the answer true.

Natural Numbers

1 Addition is closed under the Natural Numbers.

	true
	false

2 Subtraction is closed under the Natural Numbers.

	true
	false

3 Multiplication is closed under the Natural Numbers.

	true
	false

4 Division is closed under the Natural Numbers.

	true
	false

5 Exponentiation is closed under the Natural Numbers.

	true
	false

6 Roots are closed under the Natural Numbers.

	true
	false

Whole Numbers

7 Addition is closed under the Whole Numbers.

	true
	false

8 Subtraction is closed under the Whole Numbers.

	true
	false

9 Multiplication is closed under the Whole Numbers.

	true
	false

10 Division is closed under the Whole Numbers.

	true
	false

11 Exponentiation is closed under the Whole Numbers.

	true
	false

12 Roots are closed under the Whole Numbers.

	true
	false

Integers

13 Addition is closed under the Integers.

	true
	false

14 Subtraction is closed under the Integers.

	true
	false

15 Multiplication is closed under the Integers.

	true
	false

16 Division is closed under the Integers.

	true
	false

17 Exponentiation is closed under the Integers.

	true
	false

18 Roots are closed under the Integers.

	true
	false

Rational Numbers

19 Addition is closed under the Rational Numbers.

	true
	false

20 Subtraction is closed under the Rational Numbers.

	true
	false

21 Multiplication is closed under the Rational Numbers.

	true
	false

22 Division is closed under the Rational Numbers.

	true
	false

23 Exponentiation is closed under the Rational Numbers.

	true
	false

24 Roots are closed under the Rational Numbers.

	true
	false

Irrational Numbers

25 Addition is closed under the Irrational Numbers.

	true
	false

26 Subtraction is closed under the Irrational Numbers.

	true
	false

27 Multiplication is closed under the Irrational Numbers.

	true
	false

28 Division is closed under the Irrational Numbers.

	true
	false

29 Exponentiation is closed under the Irrational Numbers.

	true
	false

30 Roots are closed under the Irrational Numbers.

	true
	false

Real Numbers

31 Addition is closed under the Real Numbers.

	true
	false

32 Subtraction is closed under the Real Numbers.

	true
	false

33 Multiplication is closed under the Real Numbers.

	true
	false

34 Division is closed under the Real Numbers.

	true
	false

35 Exponentiation is closed under the Real Numbers.

	true
	false

36 Roots are closed under the Real Numbers.

	true
	false