geometry

Properties, postulates, sets

Set

collection of objects

Objects

elements of sets

Empty set

contains no elements

Union

combination of elements from multiple sets

Intersection

combination of same elements from multiple sets

Commutative property (add., multi.)

changing order of terms doesn’t change answer

Associative property (add., multi.)

changing grouping of numbers doesn’t affect the answer

Distributive property (add., multi.)

sharing a multiplier with a group

Properties of equality (sub., add., multi., div.)

to do the same operation to both sides of the equation to make it equal

Property of inequality (add., multi.)

to do the same operation to both sides of an inequality to show difference or similarity

Transitive property

if a first value relates to a second, and that second value relates to a third in the exact same way, then the first and third values must also relate

Trichotomy property

for any two real numbers a and b, exactly one of the relationships has to be true a>b, a<b, a=b

Postulate

given

Coordinate system

is a system of assigning points to numbers

Ruler placement postulate

given two points P and Q of a line, the coordinate system can be placed in

such a way that the coordinate of P is 0 and the coordinate of Q is positive.

Distance postulate

for every pair of points, there corresponds a unique positive number

Ruler postulate

the points of a line can be placed in correspondence with the real numbers in

such a way that

(1) to every point of the line there corresponds exactly one real number

(2) to every real number there corresponds exactly one point of the line

(3) the distance between any two points is the absolute value of the

difference of their corresponding numbers

Line postulate

for every pair of different points there is exactly one line that contains both

points