Set
group of numbers, points, values, items
Notation: {...}
Element or member
anything in the set
Subset
A is a subset of B is every element of A is also an element of B
Notation: A C B ("C" is stretched out)
Equal sets
two or more sets with exactly the same members
Notation: A=B
Intersection of sets A and B
all elements of both A and B
Notation: A [upside down U] B= {all x such that x is an element of A and B}
Union of sets A and B
all elements in A plus elements in B
Notation: A U B= {all x such that x [stretched out C with line in middle] (means it is an element) A or x is an element of B}
Disjoint sets
two or more sets with no elements in common
Empty set
-set with no elements
-subset of all
Notation: { }, null set sign
Zero set
set with only one element->0
Counting/Natural numbers
-the numbers we use when counting
-1,2,3,4...
Notation: N
Whole numbers
-natural numbers and zero
-0,1,2,3...
Notation: W
Integers
-all whole numbers and their opposites
-...-4, -3, -2, -1, 0, 1, 2, 3, 4...
Notation: Z
Rational numbers
-any number that can be expressed as a ratio/fraction of two integers
-decimal that terminates, integer, decimal that repeats
-EX: 2, 0.5, 0.33333...
Notation: Q
Irrational numbers
-any number that cannot be expressed as a ratio/fraction of two integers
-decimales that don't terminate or repeat
-EX: pi, square root of 2
Real numbers
-the union of rational and irrational numbers
Notation: R
Complex numbers
any number that can be written as "a+bi; all real numbers and imaginary numbers
Commutative Property
*a+b=b+a
*a x b=b x a
*changing order
Associative property
*(a+b)+c=a+(b+c)
*Changing grouping
Distributive Property
a(b+c)=ab+cb
Trichotomy
For all real numbers x and y, exactly one of the following is true:
*x=y
*x>y
*x<y
Reflexive Property
a=a
Symmetric Property
If a=b, then b=a
Transitive property
Equality:
If a=b and b=c, then a=c
Inequality:
If a>b and b>c, then a>c
Addition property
Equality:
If a=b, then a+c=b+c
Inequality:
If a>b and c>d, then a+c>b+d
Multiplication Property
Equality:
If a=b, then a x c=b x c
Inequality:
If a>b and c>0, then a x c> b x c
Postulate
statement that is accepted to be true without proof; should be simple, obvious, doesn't feel like it needs to be proven
Theorems
statements that are proven to be true
Distance Postulate
to every pair of points, there corresponds a unique, positive number, which we refer to as the distance between the points
Notation: AB
Ruler Postulate
The points of a line can be placed in correspondence with the real nibers in such a way that:
1) Every point on the line corresponds to exactly one real number
2) Every real number corresponds to exactly one point
3) Give two points with the coordinates a
Rule Placement Postulate
Given two points P and Q, we can choose a coordinate system such that
-P is at zero
-Q had a positive coordinate
Inductive reasoning
-specific examples lead to generalization
-drowing a probable conclusion from a set of examples
-analyzing observations to reach a common pattern or trend
-conclusion is probably true, but not necessarily
-best guess
Deductive reasoning
-generalization leads to specific examples
-arriving at a conclusion that is based on accepted facts (laws, definitons, theorems, postulates, authority figures, etc.)
-conclusion must be true if given info is true/accurate
Conjecture
-hypothesis
-educated guess
Counterexample
-something that disproves something else
-promotes doubt
Definitions
-must be stated clearly and exactly
-narrow enough to exclude what should be excluded
-broad enough to include what should be included
-use previously defined terms (no circular terms)
Three undefined terms of geometry
Point, line, and plane
Point
-no size/area
-location
-named with capital letters
Line
-no thickness
-contains infinite amount of points
-named with lower case letter or with two points on the line
Plane
-flat surface
-no thickness
-no edges--continues without ending
-named with a capital letter or with 3 points that are non-collinear
Betweenness of points
B is between A and C if:
1) A,B, and C are different points of the same line
2) AB+BC=AC
Notation: When B is between A and C, write A-B-C or C-B-A
Theorem: Let A, B and C be points of a line with coordinates x, y, and z, respectively, if x<y<z, then A-B-C
Opposite rays
When A is between B and C, with rays AB and AC.
**Collinear rays that share an endpoint and only that one point
Ray
Let A and B be points. The ray AB is the union of:
1) segment AB
2) the set of all points C for which A-B-C. The point A is called the end points of ray ABLi
Line Postulate
For every two different points, there is exactly one line that contains both points
Point-Plotting Theorem
Let segment AB be a ray and let x be a positive number. Then there is exactly one point P of ray AB such that AP=x
Midpoint
A point M is called a midpoint of a segment AB if M is between A and B and AM=MB (on the segment)
Bisector
A set of points that intersects a segment at its midpoint and no other point
**every segment has infinitely many bisectors
Line segment
For any two points A and B, the segment AB is the union of A and B and all points C that are between A and B