Here is a clear, simple, and exam-oriented explanation of Boolean Sub-Algebra—perfect for BCA/MCA/engineering mathematics and discrete structures exams.

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⭐ Boolean Sub-Algebra – Introduction

A Boolean sub-algebra is a subset of a Boolean algebra that is itself a Boolean algebra under the same operations (AND, OR, NOT).

If ( (B, +, \cdot, ^{\prime}, 0, 1) ) is a Boolean algebra,
then a non-empty subset ( S \subseteq B ) is a Boolean sub-algebra if:

1. (S) is closed under join (OR, +)
2. (S) is closed under meet (AND, ·)
3. (S) is closed under complement (′)
4. (S) contains the identity elements 0 and 1

If these hold, then ( S ) becomes a Boolean algebra by itself.

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⭐ Formal Definition

A subset ( S \subseteq B ) is a Boolean sub-algebra if:

[
a, b \in S \Rightarrow (a + b) \in S,\ (a \cdot b) \in S,\ a^{\prime} \in S
]

and

[
0 \in S,\ 1 \in S
]

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⭐ Important Properties of Boolean Sub-algebras

1. Every Boolean sub-algebra contains 0 and 1.
2. Boolean sub-algebra must be closed under complement.
3. If (a \in S), then both (a’) and (a” = a) must be in S.
4. If (a, b \in S), then:
   • (a + b \in S)
   • (a \cdot b \in S)

Thus it automatically satisfies all Boolean laws (commutative, distributive, identity, absorption, etc.).

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⭐ Examples of Boolean Sub-Algebras

✔ Example 1: Power set Boolean Algebra

Let
[
B = \mathcal{P}(X)
]
(the power set of X)
with operations:

• OR = union
• AND = intersection
• NOT = complement

Let
[
X = {1,2,3}
]
B = all 8 subsets.

Take subset:

[
S = {\emptyset, X}
]

This set contains:

• 0 = ∅
• 1 = X
• ∅′ = X
• X′ = ∅

Closed under union and intersection.

👉 So S is a Boolean sub-algebra of B.

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✔ Example 2:

Let
[
B = \mathcal{P}({1,2,3,4})
]

Take
[
S = {\emptyset, {1,2},{3,4}, {1,2,3,4}}
]

Check:

• Complement of {1,2} = {3,4} ∈ S
• Complement of {3,4} = {1,2} ∈ S
• Intersection and union stay inside S

👉 So S is a Boolean sub-algebra.

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✔ Example 3: {0, 1}

In ANY Boolean algebra,
[
S = {0, 1}
]
is always the smallest Boolean sub-algebra.

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⭐ Non-Examples (NOT Sub-Algebras)

✘ Example 1

S = {0, a, 1} (missing complement)
If (a)’s complement is not in S → Not a sub-algebra.

✘ Example 2

S not closed under union or intersection.

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⭐ Generating a Boolean Sub-Algebra

Given an element (a) in B, the smallest Boolean sub-algebra containing a is:

[
{0, 1, a, a’}
]

This is called the Boolean sub-algebra generated by a.

It is the building block of Boolean algebra theory.

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⭐ Quick Exam Summary

Definition:

A Boolean sub-algebra is a non-empty subset S of a Boolean algebra B such that:

[
a,b \in S \Rightarrow a+b \in S,\ ab \in S,\ a’,b’ \in S
]

and it contains 0 and 1.

Examples:

• {0,1}
• {∅, X}
• {∅, A, A′, X}

Properties:

• Closed under OR, AND, NOT
• Must contain 0 and 1
• Smallest sub-algebra = {0,1}
• Sub-algebra generated by a = {0,1,a,a′}

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