Algebra/Closure/Closure/Answers

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Algebra

	Answers For Closure

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

__True__ Addition is closed under the Natural Numbers.
_False__ Subtraction is closed under the Natural Numbers.

           $1-2$

__True__ Multiplication is closed under the Natural Numbers.
_False__ Division is closed under the Natural Numbers.

           ${\frac {1}{2}}$

__True__ Exponentiation is closed under the Natural Numbers.
_False__ Roots are closed under the Natural Numbers.

           ${\sqrt 2}$

Whole Numbers

__True__ Addition is closed under the Whole Numbers.
_False__ Subtraction is closed under the Whole Numbers.

           $1-2$

__True__ Multiplication is closed under the Whole Numbers.
_False__ Division is closed under the Whole Numbers.

           ${\frac {1}{2}}$

__True__ Exponentiation is closed under the Whole Numbers.
_False__ Roots are closed under the Whole Numbers.

           ${\sqrt 2}$

Integers

__True__ Addition is closed under the Integers.
__True__ Subtraction is closed under the Integers.
__True__ Multiplication is closed under the Integers.
_False__ Division is closed under the Integers.

           ${\frac {1}{2}}$

__False__ Exponentiation is closed under the Integers.

           $2^{-1}$

_False__ Roots are closed under the Integers.

           ${\sqrt 2}$

Rational Numbers

__True__ Addition is closed under the Rational Numbers.
__True__ Subtraction is closed under the Rational Numbers.
__True__ Multiplication is closed under the Rational Numbers.
__True__ Division is closed under the Rational Numbers.
_False__ Roots are closed under the Rational Numbers.

          Using fractions blurs the line between exponents and roots.
           ${\sqrt {2}}=2^{\frac {1}{2}}$

Real Numbers

__True__ Addition is closed under the Real Numbers.
__True__ Subtraction is closed under the Real Numbers.
__True__ Multiplication is closed under the Real Numbers.
__True__ Division is closed under the Real Numbers.
_False__ Roots are closed under the Real Numbers.

          Using fractions blurs the line between exponents and roots.
           ${\sqrt {-}}1=-1^{\frac {1}{2}}$

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