Relations, Partial order

1. MODERN ALGEBRA
-K.Anitha.M.Sc.,M.Phil.,
-M.Mohana Malar.M.Sc.,M.Phil.,

2. Equivalence Relation
A relation R defined on a non empty set S is
said to be equivalence relation if it is reflexive,
symmetric and transitive.
Eg:
S=Z, a R b means ab is a perfect square.
S=R, a R b means a-b is an integer.
S=R, a R b means |a|= |b|.

3. Partial order
A relation R defined on a set S is
said to be antisymmetric if a R b and b R a
implies a = b. A relation R in S which is reflexive,
antisymmetric and transitive is called a partial
order on S.
A set S with a partial order R
defined on it is called a partially ordered or a
poset and is denoted by ( S, R).

4. Functions
Consider the function f: R Rgiven by
f(x)= 3 is called a constant function.
A function f: A B is one-one function if
distinct elements are in A have distinct image in
B under f.
A function f: A B is onto function if the
range of f is equal to B.

5. Binary operations
Eg:
The usual addition,+ is a binary
operation on N,Z,Q,R and C.
In any non empty set A,* defined by
a*b =a is a binary operation.
In( N.*) defined by a*b=aab is a binary
operations.

6. Permutation Groups
Results
Any permutation can be expressed as a product of disjoint
cycle.
Any permutations can be expressed as a product of
transpositions.
The product of two even permutations is an even
permutations.
The product of two odd permutations is an even
permutations.
The product of an even permutations and an odd
permutations is an odd permutations.
The inverse of an even permutations is an even
permutations.
The inverse of an odd permutations is an odd permutations.
The identity permutations e is an even permutations.

7. Cyclic Groups and Order of
an element.

8. Cosets and Lagrange’s
Theorem

10. Normal Subgroups and
Quotient Groups.