Here is a clear, simple, and exam-oriented introduction to Monoids and Groups — perfect for BCA/MCA/Engineering/Discrete Mathematics exams.

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⭐ Monoids and Groups – Introduction

Monoids and Groups are fundamental algebraic structures used in:

✔ Mathematics
✔ Theoretical computer science
✔ Automata theory
✔ Cryptography
✔ Data structures
✔ Abstract algebra

They help study operations on sets with certain properties.

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⭐ 1. MONOIDS

A monoid is an algebraic structure consisting of:

1. A set M
2. A binary operation (*) (closed on M)
3. An identity element in M

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⭐ Definition of a Monoid

A set ( M ) with a binary operation (*) is a monoid if:

1️⃣ Closure

For all (a, b \in M):
[
a * b \in M
]

2️⃣ Associativity

For all (a, b, c \in M):
[
(a * b) * c = a * (b * c)
]

3️⃣ Identity Element

There exists an element ( e \in M ) such that for all ( a \in M ):

[
e * a = a * e = a
]

Such (e) is called the identity of the monoid.

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⭐ Examples of Monoids

✔ Example 1: Natural numbers under addition

Set: ( \mathbb{N} = {0, 1, 2, \ldots} )
Operation: +
Identity: 0

→ (ℕ, +) is a monoid.

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✔ Example 2: Strings under concatenation

Set: all strings over alphabet Σ
Operation: concatenation
Identity: empty string ε

→ (Σ*, concatenation) is a monoid.
(Important in automata theory)

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✔ Example 3: Integers under multiplication

Set: ℤ
Operation: ×
Identity: 1

→ (ℤ, ×) is a monoid.

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⭐ Monoid vs Semigroup

Structure	Properties
Semigroup	Closure + associativity
Monoid	Semigroup + identity

So every monoid is a semigroup.

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⭐ 2. GROUPS

A group is a monoid with the additional property that every element has an inverse.

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⭐ Definition of a Group

A set ( G ) with a binary operation (*) is called a group if:

1️⃣ Closure

[
a * b \in G
]

2️⃣ Associativity

[
(a * b) * c = a * (b * c)
]

3️⃣ Identity Element

There exists ( e \in G ) such that:
[
e * a = a * e = a
]

4️⃣ Inverses

For each ( a \in G ), there exists a unique ( a^{-1} \in G ) such that:
[
a * a^{-1} = a^{-1} * a = e
]

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⭐ If the operation is also commutative:

[
a * b = b * a
]

then the group is called an Abelian (or commutative) group.

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⭐ Examples of Groups

✔ Example 1: Integers under addition

Set: ℤ
Operation: +
Identity: 0
Inverse: −a

→ (ℤ, +) is an abelian group.

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✔ Example 2: Non-zero real numbers under multiplication

Set: ℝ − {0}
Operation: ×
Identity: 1
Inverse: (a^{-1})

→ A group (abelian).

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✔ Example 3: Symmetric group ( S_n )

Set: all permutations of n objects
Operation: composition of functions
→ non-abelian group.

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⭐ Examples that are NOT Groups

✘ Natural numbers under addition (ℕ, +)

No inverse element.
→ Only a monoid, not a group.

✘ Integers under multiplication (ℤ, ×)

Except ±1, others have no inverse.
→ Monoid, not a group.

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⭐ Difference Between Monoid and Group

Feature	Monoid	Group
Closure	✔	✔
Associativity	✔	✔
Identity	✔	✔
Inverse	✘ Not required	✔ Required
Example	(ℕ, +)	(ℤ, +)

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⭐ Applications of Monoids & Groups

✔ In Computer Science

• Automata theory (Σ*, concatenation: monoid)
• State transitions
• Cryptography (group theory)
• Symmetry operations in graphics
• Error-correcting codes
• Hashing
• Permutations in algorithms

✔ In Mathematics

• Symmetry analysis
• Geometry
• Algebraic structures
• Number theory

✔ In Physics

• Particle symmetry
• Conservation laws
• Rotations and transformations (Lie groups)

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⭐ Exam-Oriented Summary

Monoid

A set with: closure + associativity + identity.

Group

Monoid + every element has an inverse.

Abelian Group

Group + commutativity.

Examples:

• (ℤ, +) → abelian group
• (ℕ, +) → monoid
• (Σ*, concatenation) → monoid
• (ℝ {0}, ×) → abelian group

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