Math 245 Exam 3

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Math 245 Exam 3
2014-04-15 04:32:13

exam 3
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  1. a mod n
    Fix a (positive) integer n. If a (element of) Z, then a mod n = remainder upon division by n (# between 0 and n-1)
  2. congruent mod n
    Fix n (element of) Z+. Two numbers a and b are congruent mod n denoted a = b(mod n) if a (divides) [b-a].
  3. modulus
    Fix a positive integer n (the modulus) for a,b (element of) Z: a +n b: = remainder of a+b upon division by n. a xn b: = remainder of axb upon division by n.
  4. permutation
    A permutation of a set of distinct objects x1,...,xn is an ordering of the objects where each object appears exactly once.
  5. r-permutation
    An r-permutation of a set of n distinct objects is an ordering of r objects out of n where each object appears at most once.
  6. r-combination
    An r-combination of a set of n distinct objects is any collection of r objects independent of order.
  7. multinomial coefficient
    Let n be a natural number and let i,...ik be natural numbers s.t. n = i1+i2+...+ik. The multinomial coefficient, denoted n (choose) i1, i2,...,ik = n! (choose) i1!i2!...ik!
  8. relation from A to B
    Let A&B be a set. A relation from A to B is a subset R (subset) AxB. R: set of ordered pairs (a,b) (element) R; denoted aRb.
  9. domain
    Given R (subset) AxB, the domain of R is the set domR = {a (element) A s.t. the exists b (element) B with (a,b) (element) R} (subset) A.
  10. range
    The range of R is the set ranR = {b (element) B s.t. there exists a (element) A with (a,b) (element) R} (subset) B.
  11. relation on
    Let A be a set. A relation on A is a relation R (subset) AxA. A relation on A is Reflexive if for all a (element) A, aRa; Symmetric if aRb, then bRa; Transitive if aRb, and bRc, then aRc; Anti-Symmetric if aRb and bRa, then a=b
  12. equivalence relation
    Let A be a non-empty set. An equivalence relation ~ on A is a relation that is reflexive, symmetric, and transitive
  13. equivalence class
    Let ~ be an equivalence relation on A. The equivalence class of a (element) A is the set a(bar) = [a] = {x (element) A s.t. x~a}.
  14. partition
    Let S be a set. A partition of S is a collection of sets {Slambda}lambda(element)LAMBDA with: Slambda (not equal) empty set for all lambda (element) LAMBDA; Slambda (intersect) Smu = empty set or Slambda = Smu; (union)lambda(element)LAMBDA Slambda = S
  15. quotient space
    Let A be a set, ~ an equivalence relation. The quotient space of A modulo ~ is the set of equivalence class in A, denoted A/~.