Axioms and definitions

associative axiom for addition

(x+y)+z=x+(y+z)

associative axiom for multiplication

(xy)z=x(yz)

multiplication property of -1

-1x= -x

multiplication property of 0

0x=0

multiplicative indentity axiom

1x=x

property

a mathematical system that is true concerning that system

inverse axioms

additive or multiplicative inverses for real numbers [x+(-x)=0 or x(1/x)+1]

11 field axioms

closure axioms, inverse axioms, indetity axioms, associative axioms, commutative axioms, and distributive axioms

associative axioms

equivalent expression where order isn't changed, but grouping is [(x+y)+z=x+(y+z) or (xy)z=x(yz)

transitive axiom of equality

if x=y and y=z then x=z because it equals y and y also equals z

addition property of equality

if x=y then x+z=y+z because x and y are the same and they are both adding the same amount (z) making them equal

multiplication property of equality

if x=y then xz=yz because since x and y equal eachother and they are being multiplied by the same amount they become equal

symmetric axiom of equality

if x=y then y=x (x=y and y=x are the same thing except x and y are flipped)

distibutive axioms

multiplication distributes over addition [x(y+z)=xy+xz]

commutative axioms

order is changed, but still equal to eachother (x+y=y+x or xy=yx)

axiom

propertry that forms a mathematical system and is assumed to be correct without proof

closure axioms

set of real numbers closed under addition or multiplication (x+y or xy)

indentity axioms

the real number keeps it identity or stays the same (x+0=x or 1x=x)

3 equality axioms

transitive axiom, symmetric axiom, and reflexive axiom

multiplicative inverses for axiom

x(1/x)=1

distributive axiom for multiplication over addition

x(y+z)=xy+xz

additive inverses axiom

x+(-x)=0

additive identity axiom

x+0=x

commutative axiom for addition

x+y=y+x

reflexive axiom of equality

x=x (a number will always be the same number whereever it is)

commutative axiom for multiplication

xy=yx