Day 06Part 04 – Discrete Mathematics – Groupoid , Semi group , monoid , Group and abelian group

Day 06Part 04 – Discrete Mathematics – Groupoid , Semi group , monoid , Group and abelian group

https://www.gyanodhan.com/video/7B4.%20GATE%20CSEIT/Discrete%20Mathematics%201/395.%20Day%2006Part%2004%20-%20Discrete%20Mathematics%20-%20Groupoid%20%2CSemi%20group%2C%20monoid%2C%20Group%20and%20abelian%20group.mp4

 Discrete Mathematics – Groupoid, Semigroup, Monoid, Group & Abelian Group

In Discrete Mathematics & Abstract Algebra, different algebraic structures help in understanding mathematical operations and their properties. Let’s go step by step!

 Groupoid

A groupoid is a non-empty set with a binary operation (a rule that combines two elements to form another element in the set).

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

$$\forall a,b\in G,a\ast b\text{ is also in }G.$$

Example: The set of natural numbers (N, +) is a groupoid since the sum of any two natural numbers is also a natural number.

 Semigroup

A semigroup is a groupoid with the associative property.

Definition: A set S with a binary operation (*) is a semigroup if:

$$\forall a,b,c\in S,(a\ast b)\ast c=a\ast (b\ast c).$$

Example: The set of natural numbers (N, +) is a semigroup because addition is associative:

$$(2+3)+4=2+(3+4).$$

 Monoid

A monoid is a semigroup with an identity element.

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

1. Associativity: $(a\ast b)\ast c=a\ast (b\ast c)$ for all $a,b,c\in M$.
2. Existence of an Identity Element (e): There exists an element $e\in M$ such that $$a\ast e=e\ast a=a,\forall a\in M.$$

Example: The set of natural numbers (N, ×) is a monoid because multiplication is associative and has an identity element 1.

$$2\times (3\times 4)=(2\times 3)\times 4,2\times 1=2.$$

 Group

A group is a monoid in which every element has an inverse.

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

1. Associativity: $(a\ast b)\ast c=a\ast (b\ast c)$, for all $a,b,c\in G$.
2. Identity Element (e): There exists an element $e\in G$ such that $$a\ast e=e\ast a=a,\forall a\in G.$$
3. Inverse Element: For every $a\in G$, there exists an element a−1a^{-1}a−1 such that a∗a−1=a−1∗a=e.a * a^{-1} = a^{-1} * a = e.a∗a−1=a−1∗a=e.

Example: The set of integers (Z, +) forms a group because:

• It is associative.
• The identity element is 0.
• Each element has an inverse: $5+(-5)=0$.

However, (N, +) is not a group because natural numbers do not have an inverse in N (e.g., 5 has no negative number in N to make the sum 0).

 Abelian Group (Commutative Group)

A group is called an Abelian (or commutative) group if the binary operation is also commutative.

Definition: A group G is Abelian if:

$$a\ast b=b\ast a,\forall a,b\in G.$$

Example: The set of integers (Z, +) is an Abelian group because addition is commutative:

$$2+3=3+2.$$

However, (Matrix multiplication) is not always commutative, so the set of matrices under multiplication does not always form an Abelian group.

 Summary of Algebraic Structures

Structure	Associativity	Identity Element	Inverse Element	Commutativity
Groupoid	(Not necessarily)
Semigroup
Monoid
Group
Abelian Group

Would you like examples or proofs for any of these concepts? Let me know!

Day 06Part 04 – Discrete Mathematics – Groupoid , Semi group , monoid , Group and abelian group

Semigroup, monoid and group models of groupoid identities

Section I.1. Semigroups, Monoids, and Groups

Discrete Mathematical Structures Unit-1

Here is a detailed explanation of Groupoid, Semigroup, Monoid, Group, and Abelian Group in Discrete Mathematics (Day 06 Part 04) — useful for GATE / UGC NET / B.Sc / B.Tech CSE / Competitive Exams.

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Algebraic Structures in Discrete Mathematics

Based on:

A set S and a binary operation * defined on it (i.e., ∗:S×S→S* : S \times S \rightarrow S)

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1⃣ Groupoid (मात्र एक बाइनरी ऑपरेशन)

Definition:

A groupoid is a non-empty set with a binary operation.
No other conditions (like associativity or identity) are required.

Example:

Set of integers ℤ with subtraction (−):
(a−b)(a – b) ∈ ℤ ⇒ it forms a groupoid.

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2⃣ Semigroup (Associative होना चाहिए)

Definition:

A semigroup is a groupoid in which the operation is associative.
i.e., (a∗b)∗c=a∗(b∗c)(a * b) * c = a * (b * c) for all a,b,c∈Sa, b, c \in S

Example:

Set of natural numbers ℕ under addition:
(a+b)+c=a+(b+c)(a + b) + c = a + (b + c)

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3⃣ Monoid (Associative + Identity Element)

Definition:

A monoid is a semigroup with an identity element ee, such that
a∗e=e∗a=aa * e = e * a = a for all a∈Sa \in S

Example:

ℕ ∪ {0} under addition: Identity = 0
Because a+0=0+a=aa + 0 = 0 + a = a

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4⃣ Group (Monoid + Inverse)

Definition:

A group is a monoid in which every element has an inverse
i.e., for every a∈Sa \in S, there exists a−1∈Sa^{-1} \in S such that
a∗a−1=a−1∗a=ea * a^{-1} = a^{-1} * a = e

Example:

ℤ under addition

• Identity = 0
• Inverse of a = –a

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5⃣ Abelian Group (Commutative Group)

Definition:

A group in which the operation is commutative
i.e., a∗b=b∗aa * b = b * a for all a,b∈Sa, b \in S

Example:

ℝ (real numbers) under addition
→ It satisfies:

• Associativity
• Identity (0)
• Inverse (–a)
• Commutativity

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Summary Table:

Structure	Closure	Associative	Identity	Inverse	Commutative
Groupoid
Semigroup
Monoid
Group
Abelian Group

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Helpful Tips:

Every Abelian Group is a Group, but not vice versa.
If an algebraic structure satisfies more properties, it’s stronger.
Use real-world examples like clock arithmetic (modulo math) for better understanding.

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Day 06Part 04 – Discrete Mathematics – Groupoid , Semi group , monoid , Group and abelian group