CSET 211: Algebraic Structures

  1. closure property
    a set is closed if and only if the operation on two elements of the set produces another element of the set
  2. commutative property
    the order of elements does not change the outcome of the operation
  3. associative property
    grouping without changing order
  4. distributive property
    describes the interaction of the two operations of multiplication and addition
  5. identity property
    the existence of a special element in a set that goes along with a particular operation 

    • 0 for addition
    • 1 for multiplication
  6. 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
  7. group
    a set of elements with one operation

    a set and operation that satisfies the properties: closure, associative, identity and inverse
  8. a∈S
    a is an element of set S
  9. refers to an operation
  10. abelian
    A group is said to be abelian if x⋅y=y⋅x for every x,y∈G [Commutative Property]
  11. matrices
    form a group under addition, multiplication and transposition
  12. ring
    consists of a set with two operations so that the set with the operation of addition forms an Abelian group
  13. 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
  14. Boolean field
    the set containing only these elements that every field must contain
  15. 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.
  16. real number
    may be either rational or irrational
  17. inequality
    a statement about the relative size or order of two objects
  18. 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
  19. 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
  20. 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
  21. 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
  22. 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
  23. rational number
    a number that can be expressed as a fraction ab where a and b are integers and b≠0
  24. 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.
Card Set
CSET 211: Algebraic Structures
Math CSET Subtest I: Algebra