# CSET 211: Algebraic Structures

 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 addition1 for multiplication inverse property 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 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 • refers to an operation 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+cif 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: aa>b transitive for inequalities For all real numbers, a, b, and c,(a)  if aa>b and  c">b>c, then  c ">a>c reversal for inequalities For real numbers,a and b,(a)  if  b">a>b, then bb>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 ac>0 and  b">a>b, then  bc">ac>bc and  b/c">ac>bc(b)  If  0">c>0 and aa>b, then acac>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. Authorcornpops ID301062 Card SetCSET 211: Algebraic Structures DescriptionMath CSET Subtest I: Algebra Updated2015-04-19T03:17:34Z Show Answers