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Day 06Part 02- Discrete Mathematics for Gate Computer – Algebraic Structure and Binary operations.

Ali Seron Jason

4 months ago

Day 06Part 02- Discrete Mathematics for Gate Computer – Algebraic Structure and Binary operations.

https://www.gyanodhan.com/video/7B4.%20GATE%20CSEIT/Discrete%20Mathematics%201/397.%20Day%2006Part%2002-%20Discrete%20Mathematics%20for%20Gate%20Computer%20-%20Algebraic%20Structure%20and%20Binary%20operations.mp4

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Contents

1 Day 06 Part 02 – Discrete Mathematics for GATE CSE

Day 06 Part 02 – 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:

$\times S \rightarrow S$

That means:
For any two elements $\in S$ , the result $\in S$ .

2. Examples of Binary Operations

Operation	Set	Binary?	Reason
$a + b$	Integers	Yes	Result is an integer
$a - b$	Natural Numbers	No	Result may be negative
$\cdot b \mod n$	$Zn\mathbb{Z}_n$	Yes	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	$∀a,b∈S,a∗b∈S\forall a,b \in S, a * b \in S$
Associativity	$a * (b * c) = (a * b) * c$
Commutativity	$a * b = b * a$
Identity	$a * e = a$ and $e * a = a$
Inverse	For each $a$ , $a * a^{-1} = e$

GATE-Style Sample Question:

Let $*$ be a binary operation defined on the set $S = \{0, 1, 2\}$ as
$\mod 3$ .
Is $(S, *)$ a group?

Solution:

Closure: Yes, all results in ${0,1,2\}$
Associative: Yes (mod addition is associative)
Identity: 0 (since $\mod 3 = a$ )
Inverse:
- 0 0
- 1 2
- 2 1
Commutative: Yes (Abelian group)

Answer: Yes, it’s 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

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Day 06Part 02- Discrete Mathematics for Gate Computer – Algebraic Structure and Binary operations.

Discrete Mathematics for Computer Science

Discrete Mathematical Structures

Mathematics (Discrete Structure).pdf