Field Axioms
Name the property/axiom:
In any field, a + b = b + a and a x b = b x a.
The Commutative Property/Axiom
Name 4 of the 8 conditions we check to determine whether a set is a field.
Any combination of these 8 is correct.
Name the four binary operations and circle the two we use to check for closure regarding fields.
Addition, Subtraction, Multiplication, and Division
Name the property/axiom:
The property stating that a x (b + c) = a x b + a x c.
The Distributive Property/Axiom
Give me a reason why the set of Whole Numbers is not a field.
No additive inverse.
or
No multiplicative inverse.
Name a binary operation that the set of natural numbers is NOT closed under.
Subtraction
or
Division
For addition, this element acts as the additive identity in every field.
0
Give me 3 reasons why the set of Irrational numbers is not a field.
No additive inverse/idenity.
No multiplicative inverse/idenity.
Not closed under addition or multiplication.
Name the binary operation that the set of Integers is not closed under.
Division
1/2 is not an element of the set of Integers.
Name the property/axiom:
The property where (a + b) + c = a + (b + c)
The Associative Property
Give me THE reason why the set of Integers is not a field.
It does not have the multiplicative inverse.
The set of odd integers is closed under multiplication (e.g., 3 x 5 = 15). Is the set of odd integers closed under addition?
(Answer is cannot be stolen)
False.
3+5 = 8
What is the multiplicative inverse for every element a in a field?
a-1 or 1/a
Set A = {-2, -1, -1/2, 0, 1, 1/2, 2}.
What axiom, identity or inverse element, or closure operation does set A lack that makes it not a field?
It is not closed under addition.
Is the set of rational numbers closed under division?
(Answer cannot be stolen)
False.
You cannot divide by 0.