CSET 211: Algebraic Structures

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

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

grouping without changing order

describes the interaction of the two operations of multiplication and addition

the existence of a special element in a set that goes along with a particular operation 

• 0 for addition
• 1 for multiplication

a non-zero 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

a set of elements with one operation

a set and operation that satisfies the properties: closure, associative, identity and inverse

a is an element of set S

refers to an operation

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

form a group under addition, multiplication and transposition

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

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

the set containing only these elements that every field must contain

• 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.

may be either rational or irrational

a statement about the relative size or order of two objects

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

• 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

• For real numbers,a and b,(a) 
• if  b">a>b, then b<a(b) 
• if  a<b then  a">b>a

• 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

• 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

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

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.