
closure property
a set is closed if and only if the operation on two elements of the set produces another element of the set

commutative property
the order of elements does not change the outcome of the operation

associative property
grouping without changing order

distributive property
describes the interaction of the two operations of multiplication and addition

identity property
the existence of a special element in a set that goes along with a particular operation
 0 for addition
 1 for multiplication

inverse property
a nonzero element X, of a set has an inverse with respect to a given operation if and only if there exists another member, Y, of the same set, such that when the operation is performed the identity element is produced

group
a set of elements with one operation
a set and operation that satisfies the properties: closure, associative, identity and inverse

a∈S
a is an element of set S


abelian
A group is said to be abelian if x⋅y=y⋅x for every x,y∈G [Commutative Property]

matrices
form a group under addition, multiplication and transposition

ring
consists of a set with two operations so that the set with the operation of addition forms an Abelian group

field
a set with two operations, addition and multiplication, such that the set with the operation of addition forms an Abelian group, and the set with the operation of multiplication also forms an Abelian group
commutative rings which have no divisors of zero

Boolean field
the set containing only these elements that every field must contain

ordered field
 In mathematics, an ordered field is a field (F,+,⋅) together with a total order ≤ on F that is compatible with algebraic operations in the following sense:
 if a≤b, then a+c≤b+c
 if 0≤a and 0≤b, then 0≤ a+b.

real number
may be either rational or irrational

inequality
a statement about the relative size or order of two objects

trichotomy for inequalities
For any real numbers, a and b, exactly one of the following is true: a<b,a=b, or b">a>b

transitive for inequalities
 For all real numbers, a, b, and c,(a) if a<b and b<c, then a<c(b)
 if b">a>b and c">b>c, then c ">a>c

reversal for inequalities
 For real numbers,a and b,(a)
 if b">a>b, then b<a(b)
 if a<b then a">b>a

addition and subtraction for inequalities
 For any real numbers, a, b, and c, (a)
 if b">a>b, then b + c">a+c>b+c and b – c">a–c>b–c(b)
 if a<b, then a+c<b+c and a–c<b–c

multiplication and division for inequalities
 For any real numbers, a, b, and c(a)
 If 0">c>0 and b">a>b, then bc">ac>bc and b/c">ac>bc(b)
 If 0">c>0 and a<b, then ac<bc and ac<bc(c)
 If c<0 and b">a>b, then ac<bc and ac<bc(d)
 If c<0 and a<b, then bc">ac>bc and b/c">ac>bc

rational number
a number that can be expressed as a fraction ab where a and b are integers and b≠0

complex number
A complex number is a number of the form a + bi where a and b are real numbers, and i is the imaginary unit, with the property i 2 = 1. The real number a is called the real part of the complex number, and the real number b is the imaginary part. When the imaginary part b is 0, the complex number is just the real number a. Complex numbers can be added, subtracted, multiplied, and divided like real numbers, but they have additional properties.

