Distributive Law in Boolean Algebra

According to the Distributive Law in Boolean Algebra, one Boolean operation can be distributed over another operation without changing the result. The Distributive Law applies to both AND and OR operations.

• In the AND operation, AND can be distributed over OR.
• In the OR operation, OR can be distributed over AND.

Distributive laws are fundamental laws used to simplify Boolean expressions and understand the behavior of digital logic circuits.

Types of Distributive Laws

There are two types of Distributive Laws in Boolean algebra:

1. AND Distributive Law

The Distributive Law for the AND operation states that AND can be distributed over an OR expression. It can be expressed as:

A · (B + C) = (A · B) + (A · C)

The following diagram of the AND Distributive Law explains it.

Let’s explain the diagram.

A · (B + C) = (A · B) + (A · C). It means that A is ANDed with the OR operation of B and C, which gives the same output as ANDing A separately with B and C and then ORing the results.

• If A = 1, B = 0, and C = 1, then 1 · (0 + 1) = 1 and (1 · 0) + (1 · 1) = 1.
• If A = 0, B = 1, and C = 1, then 0 · (1 + 1) = 0 and (0 · 1) + (0 · 1) = 0.

Therefore, A · (B + C) = (A · B) + (A · C) because both expressions produce the same output. It is called the AND Distributive Law.

2. OR Distributive Law

The Distributive Law for the OR operation states that OR can be distributed over an AND expression. It can be expressed as:

A + (B · C) = (A + B) · (A + C)

The following diagram of the OR Distributive Law explains it.

Let’s explain the diagram.

A + (B · C) = (A + B) · (A + C). It means that A is ORed with the AND operation of B and C, which gives the same output as ORing A separately with B and C and then ANDing the results.

• If A = 0, B = 1, and C = 1, then 0 + (1 · 1) = 1 and (0 + 1) · (0 + 1) = 1.
• If A = 1, B = 0, and C = 1, then 1 + (0 · 1) = 1 and (1 + 0) · (1 + 1) = 1.

Therefore, A + (B · C) = (A + B) · (A + C) because both expressions produce the same output. It is called the OR Distributive Law.

Distributive Law Truth Table

Let’s explain the truth table of AND and OR Distributive Laws in Boolean algebra.

AND Distributive Law Truth Table

In the AND Distributive Law, AND is distributed over OR without changing the output. The following diagram shows the truth table of the AND Distributive Law.

So, A · (B + C) = (A · B) + (A · C) because both expressions produce the same output for every combination of A, B, and C.

OR Distributive Law Truth Table

In the OR Distributive Law, OR is distributed over AND without changing the output. The following diagram shows the truth table of the OR Distributive Law.

So, A + (B · C) = (A + B) · (A + C) because both expressions produce the same output for every combination of A, B, and C.

Distributive Law Logic Gates

Let’s explain the logic gates of the Distributive Law in Boolean algebra.

AND Distributive Law – Logic Gate

In the AND Distributive Law, the AND operation is distributed over the OR operation. The following diagram shows the AND Distributive Law using logic gates.

The first circuit represents A · (B + C), while the second circuit represents (A · B) + (A · C). Both circuits produce the same output.

OR Distributive Law – Logic Gate

In the OR Distributive Law, the OR operation is distributed over the AND operation. The following diagram shows the OR Distributive Law using logic gates.

The first circuit represents A + (B · C), while the second circuit represents (A + B) · (A + C). Both circuits produce the same output.

Distributive Law – Circuit Switches

Let’s explain the Distributive Law using circuit switches. The Distributive Law can be represented using circuit switches to show that distributing one operation over another does not change the final output.

AND Distributive Law – Circuit Switches

In a circuit representing AND over OR, the output is 1 when A is closed and at least one of B or C is closed. Distributing A over B and C does not change the result, giving:

A · (B + C) = (A · B) + (A · C)

In the first circuit, A is combined with the parallel combination of B and C. In the second circuit, A is combined separately with B and C, and the two results are connected in parallel. Both circuits produce the same output.

• Case 01: When A = 0, B = 0, and C = 0, the output is 0.
• Case 02: When A = 0, B = 1, and C = 1, the output is 0.
• Case 03: When A = 1, B = 0, and C = 1, the output is 1.
• Case 04: When A = 1, B = 1, and C = 1, the output is 1.

Therefore, distributing A over B and C does not change the output.

OR Distributive Law – Circuit Switches

In a circuit representing OR over AND, the output is 1 when A is closed or both B and C are closed. Distributing A over B and C does not change the result, giving:

A + (B · C) = (A + B) · (A + C)

In the first circuit, A is connected in parallel with the series combination of B and C. In the second circuit, A is connected in parallel with B and C separately, and the two results are connected in series. Both circuits produce the same output.

• Case 01: When A = 0, B = 0, and C = 0, the output is 0.
• Case 02: When A = 0, B = 0, and C = 1, the output is 0.
• Case 03: When A = 0, B = 1, and C = 1, the output is 1.
• Case 04: When A = 1, B = 0, and C = 0, the output is 1.

Therefore, distributing A over B and C does not change the output.

Distributive Law in Boolean Simplification

The Distributive Law is commonly used to expand or factor Boolean expressions without changing their value. Let’s explain some examples.

Distributive Law Example 1: A · (B + C)

The AND Distributive Law allows us to distribute A over B and C.

• A · (B + C) = AB + AC

Distributive Law Example 2: A + (B · C)

The OR Distributive Law allows us to distribute A over B and C.

• A + B.C = (A + B).(A + C)

Distributive Law Example 3: A(B + C + D)

The AND operation can be distributed over multiple OR terms.

• A(B + C + D) = AB + AC + AD

Distributive Law Example 4: A + B(C + D)

The Distributive Law can be applied to the AND expression inside the OR expression.

• A + B(C + D) = A + BC + BD