Hours: 08 | Marks: 08
Group म्हणजे असा set G ज्यामध्ये operation defined असतो आणि खालील चार properties satisfy होतात:
Closure property
Associative property
Identity element
Inverse element
(Z, +) → Group
Integers under addition
Identity unique असतो
Inverse unique असतो
Definition:
Group चा subset जो स्वतः group असतो
Example:
Even integers is subgroup of integers
Definition:
Single element पासून complete group generate होतो
Example:
(Z, +)
Definition:
Generator पासून group तयार होण्यासाठी लागणारी elements संख्या
Example:
Order of group Z₆ = 6
Hours: 07 | Marks: 07
Definition:
Group चा subset
Types:
Left coset
Right coset
Statement:
Order of subgroup divides order of group
Formula:
|G| / |H|
Definition:
Left coset = Right coset
Symbol:
H ◁ G
Definition:
Group formed by cosets
Symbol:
G / H
Hours: 08 | Marks: 08
Definition:
Structure preserving function
Example:
f(x + y) = f(x) + f(y)
Output group
Definition:
Elements mapped to identity
Definition:
Two groups structurally same
Symbol:
G ≅ H
Statement:
G / Kernel ≅ Image
Hours: 07 | Marks: 07
Ring म्हणजे set ज्यामध्ये addition आणि multiplication defined असतात
Example:
(Z, +, ×)
ab = ba
Multiplicative identity exists
ab = 0
Integral domain
Subset which is ring
No zero divisor
Example:
Integers
Every element has inverse
Example:
Real numbers
Subset of field
Basic field
Example:
Rational numbers
CO1. Solve first order differential equations using different techniques.
CO2. Solve higher order differential equations.
CO3. To find the orthogonal trajectories of the curve.
CO4. Describe the different methods to solve second order differential equations.
CO5. Define the Wronskian and explain its significance in the context of linear differential
equations.
CO6. Calculate the total number of permutations for a given finite set
CO7. Apply group and Ring axioms to determine whether a set with a binary operation forms a group
and ring.
CO8. Apply the ordinary differential equation for solving various physical, chemical and daily life
problems
BSC Semester 4 Mathematics Major Paper 2 (NEP)
No Review found
