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Posted on May 13, 2026 in Computer Engineering

1. Algebraic System

An algebraic system is a non-empty set together with one or more operations (like addition or multiplication) defined on it.
Example: A set AAA with an operation ∗*∗ is written as (A,∗)(A, *)(A,∗).

2. Semigroup

A semigroup is a set SSS with a binary operation ∗*∗ that is associative:

(a∗b)∗c=a∗(b∗c),∀a,b,c∈S(a * b) * c = a * (b * c), \quad \forall a,b,c \in S(a∗b)∗c=a∗(b∗c),∀a,b,c∈S

3. Monoid

A monoid is a semigroup that has an identity element eee, such that:

a∗e=e∗a=a,∀a∈Sa * e = e * a = a, \quad \forall a \in Sa∗e=e∗a=a,∀a∈S

4. Group

A group is a monoid in which every element has an inverse.
So it satisfies:

1. Closure
2. Associativity
3. Identity element
4. Inverse element

5. Abelian Group

An Abelian group (or commutative group) is a group where the operation is commutative:

a∗b=b∗a,∀a,b∈Ga * b = b * a, \quad \forall a,b \in Ga∗b=b∗a,∀a,b∈G

6. Generation of Subgroups

If HHH is a subgroup of GGG, then generation means forming HHH from a subset A⊆GA \subseteq GA⊆G.
The subgroup generated by AAA, denoted ⟨A⟩\langle A \rangle⟨A⟩, is the smallest subgroup containing AAA.

7. Cyclic Group

A cyclic group is a group generated by a single element ggg:

G=⟨g⟩={gn∣n∈Z}G = \langle g \rangle = \{ g^n \mid n \in \mathbb{Z} \}G=⟨g⟩={gn∣n∈Z}

8. Coset

Let HHH be a subgroup of GGG and a∈Ga \in Ga∈G.

• Left coset: aH={ah∣h∈H}aH = \{ ah \mid h \in H \}aH={ah∣h∈H}
• Right coset: Ha={ha∣h∈H}Ha = \{ ha \mid h \in H \}Ha={ha∣h∈H}

9. Normal Subgroup

A subgroup NNN of GGG is normal if:

gN=Ng,∀g∈GgN = Ng, \quad \forall g \in GgN=Ng,∀g∈G

Denoted by N⊴GN \trianglelefteq GN⊴G

10. Ring

A ring is a set RRR with two operations (+ and ×) such that:

• RRR is an Abelian group under addition
• Multiplication is associative
• Distributive laws hold:

a(b+c)=ab+ac,(a+b)c=ac+bca(b+c) = ab + ac, \quad (a+b)c = ac + bca(b+c)=ab+ac,(a+b)c=ac+bc

11. Field

A field is a ring in which:

• Every non-zero element has a multiplicative inverse
• Multiplication is commutative

12. Integral Domain

An integral domain is a commutative ring with:

• No zero divisors (i.E., ab=0⇒a=0ab = 0 \Rightarrow a = 0ab=0⇒a=0 or b=0b = 0b=0)
• Multiplicative identity

Here are clear definitions followed by the proof:

Definitions

1. Tree In Graph Theory, a tree is a connected graph with no cycles

• It connects all its vertices

• It has no closed loops

2. Binary Tree

A binary tree is a hierarchical data structure in which each node has at most two children, usually called:

• left child

• right child

3. M-ary Tree An M-ary tree (or n-ary tree) is a tree in which each node has at most M children

• If ( M = 2 ), it becomes a binary tree

• Used in structures like heaps and search trees

4. Spanning Tree Given a connected graph, a spanning tree is a subgraph that:

• includes all vertices of the original graph

• is a tree (no cycles, connected)
  

• has the minimum number of edges needed to connect all vertices

5. Root In a rooted tree, the root is the topmost node from which all nodes descend

• It has no parent

• Every other node is reachable from it

Theorem

A tree with ( n ) vertices has exactly ( n – 1 ) edges.

Proof (by Induction)

Base Case

For ( n = 1 ):

• A tree with one vertex has 0 edges

• ( n – 1 = 1 – 1 = 0 ) ✔️ True

Inductive Hypothesis

Assume that any tree with ( k ) vertices has ( k – 1 ) edges.

Inductive Step

Consider a tree with ( k + 1 ) vertices.

• A fundamental property of trees:
  Every tree has at least one leaf (a vertex with degree 1)

• Remove one leaf and its edge:

  • Remaining graph has ( k ) vertices

  • By hypothesis → it has ( k – 1 ) edges

• Add back the removed vertex and edge:

  • Total edges = ( (k – 1) + 1 = k )

Thus, a tree with ( k + 1 ) vertices has ( k = (k+1 – 1) ) edges.