Day 06Part 02- Discrete Mathematics for Gate Computer β Algebraic Structure and Binary operations.
Hereβs a detailed learning guide for:
Contents
- 1 Day 06 Part 02 β Discrete Mathematics for GATE CSE
- 1.1 Topic: Algebraic Structures & Binary Operations
- 1.2 1. What is a Binary Operation?
- 1.3 2. Examples of Binary Operations
- 1.4 3. Algebraic Structures
- 1.5 4. Important Properties of Binary Operations
- 1.6 GATE-Style Sample Question:
- 1.7 Quick Revision Table:
- 1.8 Useful for GATE Topics:
- 1.9 Day 06Part 02- Discrete Mathematics for Gate Computer β Algebraic Structure and Binary operations.
- 1.10 Discrete Mathematics for Computer Science
- 1.11 Discrete Mathematical Structures
- 1.12 Mathematics (Discrete Structure).pdf
Day 06 Part 02 β Discrete Mathematics for GATE CSE
Topic: Algebraic Structures & Binary Operations
1. What is a Binary Operation?
A binary operation on a set S is a function:
β:SΓSβS*: S \times S \rightarrow S
That means:
For any two elements a,bβSa, b \in S, the result aβbβSa * b \in S.
2. Examples of Binary Operations
Operation | Set | Binary? | Reason |
---|---|---|---|
a+ba + b | Integers | Result is an integer | |
aβba β b | Natural Numbers | Result may be negative | |
aβ bmodββna \cdot b \mod n | Zn\mathbb{Z}_n | Remains in Zn\mathbb{Z}_n |
3. Algebraic Structures
An algebraic structure is a set equipped with one or more binary operations. Here are common types:
Semigroup
-
Set + Binary Operation
-
Operation is Associative
Monoid
-
Semigroup + Identity Element
Group
-
Monoid + Inverse Exists for All Elements
Abelian Group
-
Group + Commutativity
4. Important Properties of Binary Operations
Property | Definition |
---|---|
Closure | βa,bβS,aβbβS\forall a,b \in S, a * b \in S |
Associativity | aβ(bβc)=(aβb)βca * (b * c) = (a * b) * c |
Commutativity | aβb=bβaa * b = b * a |
Identity | aβe=aa * e = a and eβa=ae * a = a |
Inverse | For each aa, aβaβ1=ea * a^{-1} = e |
GATE-Style Sample Question:
Let β* be a binary operation defined on the set S={0,1,2}S = \{0, 1, 2\} as
aβb=(a+b)modββ3a * b = (a + b) \mod 3.
Is (S,β)(S, *) a group?
Solution:
-
Closure: Yes, all results in {0,1,2}\{0,1,2\}
-
Associative: Yes (mod addition is associative)
-
Identity: 0 (since a+0modββ3=aa + 0 \mod 3 = a)
-
Inverse:
-
0
0 -
1
2 -
2
1
-
-
Commutative: Yes (Abelian group)
Answer: Yes, itβs an Abelian Group
Quick Revision Table:
Structure | Associative | Identity | Inverse | Commutative |
---|---|---|---|---|
Semigroup | ||||
Monoid | ||||
Group | ||||
Abelian Group |
Useful for GATE Topics:
-
Algebraic structures
-
Group Theory
-
Boolean Algebra
-
Modular Arithmetic
Would you like:
-
A PDF notes version?
-
More GATE MCQs with solutions?
-
Visual diagram of structure hierarchy?
Just let me know!