Field Axioms 1

Explain

100

What is a field?

Any set of numbers that obeys all 11 axioms is a field

200

List all 11 Field Axioms

1/2. Closure for Addition and Multiplication

3/4.  Commutative for Addition and Multiplication

5/6.  Associative for Addition and Multiplication

7. Distributve

8/9 Multiplicative and Additive Identity

10/11 Multiplicative and Additive Inverse

300

What is the Additive Identity Element and the Multiplicative Identity Element

Additive Identity Element is 0

Multiplicative Identity Element is 1

400

Explain the Additive Inverse Axiom

x + (-x) = 0

For every real number, x, you can add the opposite to that number (-x) and the result is zero

500

Explain the multiplicative inverse axiom.

x * 1/x = 1

For every real number, x, you can multiply that number by the reciprocal, 1/x, and the result is 1

Name that Axiom

100

a + b = b + a

Commutative for Addition

200

a + (b+ c) = (a + b) + c

Associative for Addition

300

a(b +c) = ab + ac

Distributive

400

a + 0 = a

Additive Identity

500

a + (-a) = 0

Additive Inverse

Name that Axiom 2

100

ab = bc

Commutative for Multiplication

200

a(bc) = (ab) c

Associative for Multiplication

300

a * 1 = a

Multiplicative Identity

400

a * 1/a = 1

Multiplicative Inverse

500

x *(y+z) + ( - [x * (y+z)])

Additive Identity

Name that Axiom 3

100

x = x

Reflexive Property

200

If x = y, then y = x

Symmetry

300

If x = y and y = z, then x = z

Transitivity for Equality 

400

If x > y. and y > z, then x > z

If x < y and y < z, then x < z

Transitivity for Order

500

If x ad y are real numbers, then one of the following is true

y < x      OR      y > x      OR      y = x 

Trichotomy

Give an Example

100

Using variables to stand for numbers, write an example of the following axiom:

Commutative for Addition

Commutative for Multiplication

Addition:     a + b = b + a

Multiplication:  ab = ba

200

Using variables to stand for numbers, write an example of the following axiom:

Associative for Addition

Associative for Multiplication

Addition:  a + (b + c) = (a+ b) + c

Multiplication:  a(bc) = (ab)c

300

Using variables to stand for numbers, write an example of the following axiom:

Distributive

a (b + c)  = ab + ac

400

Using variables to stand for numbers, write an example of the following axiom:

Identity Element for Addition

a + 0 = a

500

Using variables to stand for numbers, write an example of the following axiom:

Identity Element for Multiplication

a * 1 = a