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

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

Here’s a detailed learning guide for:

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:

$$\ast :S\times S\to S$$

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

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
$a\cdot b\mathrm{mod}n$	Zn\mathbb{Z}_nZn​	Yes	Remains in Zn\mathbb{Z}_nZn​

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

GATE-Style Sample Question:

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

Solution:

• Closure: Yes, all results in $\{0,1,2\}$

• Associative: Yes (mod addition is associative)

• Identity: 0 (since $a+0\mathrm{mod}3=a$)

• 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