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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.

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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:

βˆ—:SΓ—Sβ†’S*: S \times S \rightarrow S

That means:
For any two elements a,b∈Sa, b \in S, the result aβˆ—b∈Sa * b \in S.


2. Examples of Binary Operations

Operation Set Binary? Reason
a+ba + b Integers Yes Result is an integer
aβˆ’ba – b Natural Numbers No Result may be negative
aβ‹…bmod  na \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

Monoid

Group

Abelian Group


4. Important Properties of Binary Operations

Property Definition
Closure βˆ€a,b∈S,aβˆ—b∈S\forall a,b \in S, a * b \in S
Associativity aβˆ—(bβˆ—c)=(aβˆ—b)βˆ—ca * (b * c) = (a * b) * c
Commutativity aβˆ—b=bβˆ—aa * b = b * a
Identity aβˆ—e=aa * e = a and eβˆ—a=ae * a = a
Inverse For each aa, aβˆ—aβˆ’1=ea * a^{-1} = e

GATE-Style Sample Question:

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

Solution:

Answer: Yes, it’s an Abelian Group


Quick Revision Table:

Structure Associative Identity Inverse Commutative
Semigroup
Monoid
Group
Abelian Group

Useful for GATE Topics:


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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