Math -11 Field Axioms/Properties
Terms
undefined, object
copy deck
- Closure Axiom of Addition
-
CLAA
If a+b=c, then c is a real number - Closure Axiom of Multiplication
-
ClAM
If ab=c, then c is a real number - Commutative Axiom of Addition
-
CAA
a+b=b+a - Commutative Axiom of Multiplication
-
CAM
ab=ba - Associative Axiom of Addition
-
AAA
(a+b)+c=a+(b+c) - Associative Axiom of Multiplication
-
AAM
(ab)c=a(bc) -
Axiom of Zero for Addition
(Identity for Addition) -
A0A
(Id+)
a+0=0+a=a -
Axiom of One for Multiplication
(Identity for Multiplication) -
A1M
(Idx)
ax1=1a=a - Axiom of Additive Inverses
-
AAI
a+(-a)=-a+a=0 - Axiom of Multiplicative Inverses
-
AMI
*A cannot equal 0*
a x 1/a = 1/a x a =1 - Distributive Axiom of Multiplication over Addition
-
DAMA
a(b+c)=ab+ac - Reflexive Axiom
- a=a
- Symmetric Axiom
- If a=b, then b=a
- Transitive Axiom
- If a=b and b=c, then a=c
- Definition of Subtraction
- a-b=a+(-b)
- Definition of Division
- a(division symbol)b or a/b = a x 1/b
- Binary Operation
- A rule for combining two real numbers (or things) to get a unique (one and only one!) real number (or thing)
- upside down A
-
means "for all, for each, for every, for any..."
universal quantifier - backwards E
- means "there exists for at least one, for some..."
- backwards E with !
- means "there is exactly one x, or a unique x"
- straight vertical line
- means "such that"
- Ring
- system in math with these axioms is called a ring