FIT1047 Chap.3 Boolean Algebra, Truth Tables and Normal Forms
Boolean Algebra, Truth Tables and Normal Forms
Boolean algebra treats zero and one as false/true states under logical operators, not ordinary arithmetic. This chapter connects four representations of one function: a natural-language requirement, complete truth table, symbolic expression and gate circuit. It teaches systematic row generation, intermediate columns, positive variable definitions and targeted tests for necessary, sufficient and exclusive conditions.
Algebraic laws—including complements, distribution, absorption and De Morgan transformations—are paired with row-set explanations instead of memorised symbol moves. Canonical sum-of-products and product-of-sums forms are derived directly from output-one and output-zero rows, providing an auditable path from a table. Equivalence requires every allowed assignment; one counterexample disproves it.
The Week 1 foundation and live Written specification make this topic especially important, but every requirement and row set here is newly authored and must not be substituted for the personalised assessment. The safest workflow is a round trip among requirement, expression, table and circuit. Each translation should preserve the same complete mapping, while intermediate columns make the first divergence observable.
Targeted cases can disprove a proposed equivalence quickly, but only full enumeration or a valid algebraic argument proves it. A1 details are used only to explain the verified skill boundary; no personalised condition, row set or assessment response is reconstructed.
What this chapter covers
- 01
Boolean variables and operator meanings
- 02
Complete truth-table enumeration
- 03
Requirement translation
- 04
Identity, complement and absorption laws
- 05
De Morgan transformations
- 06
Minterms and canonical SOP
- 07
Maxterms and canonical POS
- 08
Equivalence and targeted counterexamples
- 09
Round trips between requirement, table and expression
- 10
Intermediate columns as fault-localisation evidence
- 11
Distinguishing disproof from complete proof
AskSia-authored practice weighting (not an official mark scheme): Fresh rule-to-function example
- translateDefine positive variables S, B and E and write A=(S AND NOT B) OR E.
- tableGenerate all eight SBE rows with intermediate NOT B and S AND NOT B columns.
- canonicalTurn every output-one row into a minterm and OR them.
- simplifySimplify the canonical expression back to the direct function using labelled laws.
- verifyCheck S=1,B=1,E=0 and an emergency row to expose polarity mistakes.
Key terms
- Boolean function
- A mapping from Boolean input assignments to one Boolean output.
- Truth table
- An exhaustive list of input combinations and corresponding outputs.
- Minterm
- An AND term containing every input once and true on exactly one row.
- Maxterm
- An OR term containing every input once and false on exactly one row.
- Canonical SOP
- An OR of minterms corresponding to output-one rows.
- Equivalence
- Equality of two Boolean functions on every allowed input assignment.
- Variable legend
- A positive statement of what each Boolean symbol means before negation is applied. It anchors the translation from natural language and prevents an internally consistent table from implementing the opposite real-world condition.
- Targeted counterexample
- A carefully chosen assignment on which two proposed expressions produce different outputs. One such row disproves equivalence, but a collection of matching targeted rows cannot replace full enumeration or a valid algebraic proof.
Boolean Algebra, Truth Tables and Normal Forms FAQ
How many rows belong in a truth table?
Use every combination of independent inputs. Generate them like binary counting and check the input columns before evaluating the function.
What is the safest way to apply De Morgan’s law?
Circle the complete scope of the outer NOT, change AND↔OR, negate every enclosed operand, then verify decisive rows such as all true or one false.
How is a minterm written from a row?
Use an uncomplemented variable for a row value one and a complemented variable for row value zero, then AND all literals.
Can a few matching rows prove equivalence?
No. A few rows support examples, while equivalence requires every assignment or a valid algebra chain. One differing row is enough to disprove it.
Why keep requirement cases after algebra?
Algebra proves a final expression equals the initial expression; it does not prove the initial translation captured the intended real-world rule.
Why should intermediate truth-table columns mirror the expression tree?
They expose the value of each subexpression under every assignment. When the final output is wrong, the first divergent column identifies whether the cause is negation scope, operator choice, precedence or an earlier requirement translation.
What does a safe A1 practice boundary look like?
Use newly authored requirements and row sets to practise the verified skills. Do not enter, transform, solve or draft reflection for the personalised live task with generative AI. Follow the current Moodle instructions for the actual assessment.
Assessment move
For each fresh requirement, write a positive variable legend, identify a forbidden or decisive row and only then form the expression. Generate complete rows in one consistent order and create intermediate columns that mirror the expression tree. Practise De Morgan and absorption with both algebra and case explanations.
Build one canonical SOP from output-one rows and one canonical POS from output-zero rows, labelling every term with its source row. Simplify line by line with the law named in the margin, then create an independent truth table for the final expression. Design targeted counterexamples for inclusive/exclusive OR, negation scope and implication direction.
Keep the current Written task separate: generated assessment logic or reflection is prohibited. Build a three-view proof for every major result. First, test meaningful requirement cases, including one forbidden row and one exception. Second, enumerate the full mapping. Third, transform algebra with a law written beside each line.
If the views disagree, preserve them and locate the first differing row or law rather than editing the original requirement. Practise reading “if,” “only if,” “unless” and “exactly one” by naming the single row or cases they must reject. For minterms and maxterms, evaluate each term on its originating row and a row differing in one input before combining the full expression.
Reorder variables between two practice tables, then explicitly realign complete assignments to demonstrate why output-column counts alone cannot prove equivalence. Once weekly, invent a plausible wrong expression and design the smallest counterexample that separates it from the correct function. This develops method recognition while avoiding any reuse of the live personalised task.
Add an assessment-integrity column to the practice record: source concept, original scenario, independent derivation and verification route. The supplied text may confirm operator meanings and standard forms, but its wording, diagrams, examples and solutions remain copyrighted course material and should not be copied into a submission or study product.
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