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A binary relation R on a set X is a subset of the product X×X. Very often instead of writing, say, (x,
y)∈R we write xRy. A binary relation may have many properties; the ones of interest for the definition
of order are listed below (I'll omit the expression "for all x" or "for all y". The properties listed are
assumed to hold for all elements from X):
              Reflexivity:       xRx
              Symmetry: if xRy then yRx
              Antisymmetry:           if xRy and yRx then x = y
              Asymmetry:         if xRy then not yRx
              Transitivity:      if xRy and yRz then xRz
              Totality:     either xRy or yRx
              Density:      xRy implies existence of z such that xRz and zRy

A relation which is reflexive, symmetric and transitive is an equivalence relation. The one which is
reflexive, antisymmetric and transitive is a partial order. The one which is antisymmetric, transitive
and total is a total (or linear) order. A total order is partial because totality implies reflexivity. A set X
along with a binary relation R is said to be partially (totally) ordered if R is a partial (total) order. A
partially ordered set is called poset. (Strangely enough, a totally ordered set is not called toset.)

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   Math.Edurite.com                                                                Page : 1/3
When it comes to considering ordered sets, the partial order relation R is usually written as "≤".
Alongside a partial order "≤" it is common and convenient to work with a relation "<" defined as
x < y iff x ≤ y and x ≠ y.
The relation "<" is irreflexive, asymmetric and transitive.If A is a subset of a poset X and a ∈A is such
that, for every element b∈A, a≤b, then a is called the minimum (or the least) element of A. The
minimum element, if exists, is unique because of the antisymmetry of the partial order. A totally
ordered set in which every non-empty subset has a minimum element is called well-ordered. A finite set
with a total order is well-ordered. All total orderings of a finite set are, in a sense, the same. This is not
true of the infinite sets. The countable transfinite ordinals correspond to various well-orderings of the
set N of natural numbers. Two sets X and Y totally ordered by relations R and S are said to be similar if
they are of the same cardinality (|X| = |Y|) and there exists an order-preserving 1-1 correspondence f:
X→Y between them:            f(x)Sf(y) iff xRy.

Similar sets are said to be of the same order-type.Even without knowledge that we are working in the
multiplicative group of integers modulo n, we can show that a actually has an order by noting that the
powers of a can only take a finite number of different values modulo n, so according to the pigeonhole
principle there must be two powers, say s and t and without loss of generality s>t, such that as ≡ at
(mod n). Since a and n are coprime, this implies that a has an inverse element a-1 and we can multiply
both sides of the congruence with a-t, yielding as-t ≡ 1 (mod n).The concept of multiplicative order is a
special case of the order of group elements.

The multiplicative order of a number a modulo n is the order of a in the multiplicative group whose
elements are the residues modulo n of the numbers coprime to n, and whose group operation is
multiplication modulo n. This is the group of units of the ring Zn; it has φ(n) elements, φ being Euler's
totient function, and is denoted as U(n) or U(Zn).As a consequence of Lagrange's theorem, ordn(a)
always divides φ(n). If ordn a is actually equal to φ(n) and therefore as large as possible, then a is called
a primitive root modulo n. This means that the group U(n) is cyclic and the residue class of a generates
it.The order of a group is its cardinality, i.e., the number of elements in its set.The order, sometimes
period, of an element a of a group is the smallest positive integer m such that am = e (where e denotes
the identity element of the group, and am denotes the product of m copies of a). If no such m exists, a is
said to have infinite order. All elements of finite groups have finite order.

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    Math.Edurite.com                                                               Page : 2/3
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