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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 n1)

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) [ba].

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.

permutation
A permutation of a set of distinct objects x1,...,xn is an ordering of the objects where each object appears exactly once.

rpermutation
An rpermutation of a set of n distinct objects is an ordering of r objects out of n where each object appears at most once.

rcombination
An rcombination of a set of n distinct objects is any collection of r objects independent of order.

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!

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.

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.

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.

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; AntiSymmetric if aRb and bRa, then a=b

equivalence relation
Let A be a nonempty set. An equivalence relation ~ on A is a relation that is reflexive, symmetric, and transitive

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}.

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

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/~.

