Boolean Expressions

A boolean expression is a logical statement that evaluates to exactly one of two values: True (or 1) or False (or 0). They are the foundation of computer science, digital electronics, and algorithmic logic, used to control program flow and build circuit logic

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Building Blocks of Boolean Expressions

A Boolean expression is built of Variable, Operator and Constants.

• Variables: A, B, C, etc. (binary inputs)
• Operators: AND (·) , OR (+) , NOT (¬)
• Constants: 0 (false), 1 (true)

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Example Expression

Let’s look at a simple example:

F = A·¬B + C

Its read as Output F is 1 -> if (A AND NOT B) is True OR C is True.

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Standard Forms

Form	Structure	Output Format	Example
Sum of Products (SOP)	OR of multiple AND terms	Each AND term = 1 combination of inputs	A·B + A’·B
Product of Sums (POS)	AND of multiple OR terms	Each OR term = 1 combination of inputs	(A + B)·(A’ + B)

These forms are useful for translating expressions directly into gate level circuits.

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Canonical Forms

Canonical forms are standardized, exhaustive representations using all input variables.

Canonical Form	Components	Output value	Example
Minterms (Canonical SOP)	AND of all vars (true/false)	Equals 1	A·B·C or A’·B·C
Maxterms (Canonical POS)	OR of all vars (true/false)	Equals 0	(A + B + C’) or (A + B’ + C)

In SOP Canonical, each minterm corresponds to one row in the truth table where output is 1. You can express entire functions by listing all minterms.

Example

If output is 1 for rows:
Row 1 (A=0, B=0), Row 3 (A=1, B=0)
→ Canonical SOP: A’·B’ + A·B’

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Simplifying Expressions

Simplification helps reduce the number of gates in a circuit. This can be done using -

1. Boolean Algebra

Use boolean algebra rules to simplify expression.

Example - F = A·B + A·¬B

On applying Boolean identity: F = A(B + ¬B)

Since B + ¬B = 1, we get F = A·1 = A

This tells us the logic only depends on A and B is irrelevant.

2. Karnaugh Maps (K-maps)

A K-map is a grid-based tool to simplify Boolean expressions (best for 2-4 variables).

Steps:

1. Fill in 1s for where the output is true.
2. Group adjacent 1s in pairs, quads or octets
3. Derive a simplified term for each group (combine variables that stay constant)