Algebra/Closure/Answers

The purpose of this exercise is to get you to practice using your mathematical intuition. Examples will be given showing which answers are false. The questions that are marked True require us to define our terms. Because all of these sets of numbers have infinite members we would have to use a recursive definition. Creating an argument with a recursive definition requires a proficiency with mathematical language that you'll be ready for after reading this book.

Natural Numbers

1. __True__ Addition is closed under the Natural Numbers.
2. ________ Subtraction is closed under the Natural Numbers.
3. __True__ Multiplication is closed under the Natural Numbers.
4. ________ Division is closed under the Natural Numbers.
5. __True__ Exponentiation is closed under the Natural Numbers.
6. ________ Roots are closed under the Natural Numbers.

Whole Numbers

1. __True__ Addition is closed under the Whole Numbers.
2. ________ Subtraction is closed under the Whole Numbers.
3. __True__ Multiplication is closed under the Whole Numbers.
4. ________ Division is closed under the Whole Numbers.
5. __True__ Exponentiation is closed under the Whole Numbers.
6. ________ Roots are closed under the Whole Numbers.

Integers

1. ________ Addition is closed under the Integers.
2. ________ Subtraction is closed under the Integers.
3. ________ Multiplication is closed under the Integers.
4. ________ Division is closed under the Integers.
5. ________ Exponentiation is closed under the Integers.
6. ________ Roots are closed under the Integers.

Rational Numbers

1. ________ Addition is closed under the Rational Numbers.
2. ________ Subtraction is closed under the Rational Numbers.
3. ________ Multiplication is closed under the Rational Numbers.
4. __False Division is closed under the Rational Numbers.
5. ________ Exponentiation is closed under the Rational Numbers.
6. ________ Roots are closed under the Rational Numbers.

Irrational Numbers

1. ________ Addition is closed under the Irrational Numbers.
2. ________ Subtraction is closed under the Irrational Numbers.
3. ________ Multiplication is closed under the Irrational Numbers.
4. ________ Division is closed under the Irrational Numbers.
5. ________ Exponentiation is closed under the Irrational Numbers.
6. ________ Roots are closed under the Irrational Numbers.

Real Numbers

1. ________ Addition is closed under the Real Numbers.
2. ________ Subtraction is closed under the Real Numbers.
3. ________ Multiplication is closed under the Real Numbers.
4. _False__ Division is closed under the Real Numbers.
5. ________ Exponentiation is closed under the Real Numbers.
6. ________ Roots are closed under the Real Numbers.