ELECTENG 101 Chap.6 Boolean Algebra and Logic Gates
Boolean Algebra and Logic Gates
Truth tables, Boolean expressions and gate networks describe the same decision rule. Canonical expressions can be derived mechanically from output-one rows, then simplified with identities and checked against the original table. Reliable design keeps active-high or active-low signal meanings visible and tests boundary rows that differ by one input.
In a sum-of-products construction, every output-one row becomes an AND term containing each variable in true or complemented form, and those minterms are ORed. Boolean identities can reduce gate count and delay, but each simplification must preserve all rows rather than only a few examples. Signal names also carry meaning: a logical one might mean closed, safe or active, while an active-low input reverses the natural wording.
A final gate implementation is therefore verified against the full truth table and the stated physical interpretation.
What this chapter covers
- 01
AND, OR and NOT operations
- 02
XOR and derived gates
- 03
Boolean identities
- 04
De Morgan laws
- 05
Truth-table minterms
- 06
Expression simplification
- 07
Gate implementation
Worked example · free
Write a safety-control rule
- +2Translate the guarded path as TG.
- +2OR the override to obtain Y = TG + O.
- +3Test guarded, blocked and override rows.
Key terms
- Minterm
- A product term matching one input row where a Boolean function is true.
- Active-low signal
- A signal whose asserted physical condition is represented by logic zero.
- Universal gate
- A gate type, such as NAND or NOR, that alone can implement any Boolean function.
Boolean Algebra and Logic Gates FAQ
How do I derive an expression from a truth table?
For every output-one row, write a product containing each input in true or complemented form. OR all those minterms, then simplify. A variable appears uncomplemented when that row contains one and complemented when it contains zero. Evaluate the resulting expression on at least one output-zero row before simplifying to catch a copied bit.
Why simplify a Boolean expression?
Fewer operations can mean fewer gates, smaller delay and simpler implementation. Simplification must preserve every truth-table output. Record one identity per algebraic step so an absorption or De Morgan transformation can be audited. A shorter expression is useful only if it implements the identical input-output rule under the same active-level definitions.
Which rows should I test?
Test rows around each decision boundary, especially pairs differing in one input. All-zero and all-one rows alone rarely expose an incorrect condition. Also include the row that activates each separate product term and a nearby row that should disable it. For an override input, test both ordinary logic and the all-dominating override state.
How does active-low notation change an expression?
An active-low signal performs its named action when its electrical value is zero. Translate the physical statement into a logical predicate before writing gates, and show the complement explicitly. Do not assume a variable called ENABLE is active-high unless the notation says so; otherwise a function can be algebraically tidy but physically reversed.
When should De Morgan law be used?
Use De Morgan law when moving a complement across an AND or OR boundary: the operation changes and every input is complemented. It is useful for converting between gate families and simplifying complemented groups. Keep a bar or parentheses over the entire expression until the transformation is complete so no variable is omitted.
Why rebuild the truth table after simplification?
Algebraic slips can produce an expression that looks plausible and passes a few selected rows. Rebuilding all combinations compares the simplified rule with the specification exhaustively for a small input count. Any mismatch identifies a concrete counterexample, which is more useful than arguing from the appearance or gate count of the expression.
Exam move
Move repeatedly among words, truth tables, expressions and gates. Derive a canonical form first so correctness does not depend on spotting a clever identity. Simplify one line at a time and name the identity used. Finally rebuild a truth table from the simplified form, paying special attention to complemented and active-low variables.
Use a four-pass worksheet: define what zero and one mean, enumerate rows, derive the canonical form, and implement gates. Include an override function, an exclusive condition and an active-low safety input so the exercises demand more than memorising AND and OR symbols. During simplification, retain the unsimplified expression beside each line and annotate absorption, distributive or De Morgan steps.
Compare the original and simplified outputs row by row. End by drawing the circuit with labelled signals and estimating which gate path determines delay, linking algebraic economy to an engineering consequence without treating minimum gate count as the only goal.
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