Boolean Expressions
A Boolean expression describes a digital circuit’s behavior using binary variables (0 or 1) and logical operators. It’s like a recipe for how inputs produce an output in a circuit.
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)
Example: F = A·¬B + C , its read as Output F is 1 if (A AND NOT B) OR C is true
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.
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’
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:
- Fill in 1s for where the output is true.
- Group adjacent 1s in pairs, quads or octets
- Derive a simplified term for each group (combine variables that stay constant)
