Introduction to DLD

Associative Law in Boolean Algebra

According to the Associative Law in Boolean Algebra, changing the grouping of Boolean variables does not change the result of an operation. The Associative Law applies to both AND and OR operations.

  • In the AND operation, the grouping of variables can be changed without changing the result.
  • In the OR operation, the grouping of variables can be changed without changing the result.

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

Types of Associative Laws

There are two types of Associative Laws in Boolean algebra:

1. AND Associative Law

The Associative Law for the AND operation states that changing the grouping of variables does not change the result. It can be expressed as: (A · B) · C = A · (B · C)

The following diagram of the AND Associative Law explains it.

Associative Law in Boolean Algebra - AND Associative Law

Let’s explain the diagram.

(A · B) · C = A · (B · C). It means that changing the grouping of A, B, and C does not change the output.

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

Therefore, (A · B) · C = A · (B · C) because the output remains the same when the grouping of the variables is changed. It is called the AND Associative Law.

2. OR Associative Law

The Associative Law for the OR operation states that changing the grouping of variables does not change the result. It can be expressed as: (A + B) + C = A + (B + C)

The following diagram of the OR Associative Law explains it.

Associative Law in Boolean Algebra - OR Associative Law

Let’s explain the diagram.

(A + B) + C = A + (B + C). It means that changing the grouping of A, B, and C does not change the output.

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

Therefore, (A + B) + C = A + (B + C) because the output remains the same when the grouping of the variables is changed. It is called the OR Associative Law.

Associative Law Truth Table

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

AND Associative Law Truth Table

In an AND operation, changing the grouping of A, B, and C does not change the output. The following diagram shows the truth table of the AND Associative Law.

AND Associative Law - Truth Table

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

OR Associative Law Truth Table

In an OR operation, changing the grouping of A, B, and C does not change the output. The following diagram shows the truth table of the OR Associative Law.

OR Associative Law - Truth Table

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

Associative Law Logic Gates

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

AND Associative Law – Logic Gate

In the AND operation, the grouping of inputs can be changed without changing the output. The following diagram shows the AND Associative Law using logic gates.

AND ( Associative Law ) Logic Gate

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

OR Associative Law – Logic Gate

In the OR operation, the grouping of inputs can be changed without changing the output. The following diagram shows the OR Associative Law using logic gates.

OR ( Associative Law ) Logic Gate

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

Associative Law – Circuit Switches

Let’s explain the Associative Law using circuit switches. The Associative Law can be represented using circuit switches to show that changing the grouping of switches does not change the final output.

AND Associative Law – Circuit Switches

In a series circuit, all switches must be closed for the output to be 1. Changing the grouping of the switches does not change the result, giving:

(A · B) · C = A · (B · C)

Associative Law - AND Circuit Switches

In the first circuit, A and B are grouped together before combining with C. In the second circuit, B and C are grouped together before combining with A. 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 0.
  • Case 04: When A = 1, B = 1, and C = 1, the output is 1.

Therefore, changing the grouping of the switches does not change the output.

OR Associative Law – Circuit Switches

In a parallel circuit, the output is 1 when at least one switch is closed. Changing the grouping of the switches does not change the result, giving:

(A + B) + C = A + (B + C)

Associative Law - OR Circuit Switches

In the first circuit, A and B are grouped together before combining with C. In the second circuit, B and C are grouped together before combining with A. 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 1.
  • Case 03: When A = 0, B = 1, and C = 0, the output is 1.
  • Case 04: When A = 1, B = 1, and C = 1, the output is 1.

Therefore, changing the grouping of the switches does not change the output.

Associative Law in Boolean Simplification

The Associative Law is commonly used to change the grouping of variables in Boolean expressions without changing their value. Let’s explain some examples.

Associative Law Example 1: (A · B) · C

The AND Associative Law allows us to change the grouping of A, B, and C.

  • (A · B) · C = A · (B · C)

Associative Law Example 2: (A + B) + C

The OR Associative Law allows us to change the grouping of A, B, and C.

  • (A + B) + C = A + (B + C)

Associative Law Example 3: (AB)C

The grouping of AND terms can be changed without changing the result.

  • (AB)C = A(BC)

Associative Law Example 4: (A + B) + C

The grouping of OR terms can be changed without changing the result.

  • (A + B) + C = A + (B + C)

Associative Law Example 5: (AB)CD

The grouping of multiple AND variables can be changed without changing the result.

  • ((AB)C)D = (AB)(CD)
  • ((AB)C)D = A(B(CD))

Associative Law Example 6: (A + B) + (C + D)

The grouping of OR variables can be changed without changing the result.

  • (A + B) + (C + D) = A + (B + (C + D))
  • (A + B) + (C + D) = (A + B + C) + D

Associative Law Example 7: ABC

The AND operation can be grouped in different ways without changing the result.

  • (AB)C = A(BC)

Associative Law Example 8: A + B + C

The OR operation can also be grouped in different ways without changing the result.

  • (A + B) + C = A + (B + C)