Generated worked-example catalogue (all remaining engine examples)
The complete generated catalogue is available as data: /logic/api/dataset.json.
- Entry rule: (member OR invited) AND NOT banned, checked every way
- Entry rule slip: member OR (invited AND NOT banned) and the case that differs
- Majority of three (2-out-of-3 voting): truth table, K-map and NAND form
- Odd parity of three inputs: A XOR B XOR C
- Exactly one of three: why XOR is the wrong tool
- At most one of three: mutual exclusion
- Half adder sum: A XOR B
- Half adder carry: A AND B
- Full adder sum: A XOR B XOR Cin
- Full adder carry-out: (A AND B) OR (Cin AND (A XOR B))
- 2-to-1 multiplexer: (S AND B) OR (NOT S AND A)
- One-bit comparator: A greater than B
- Two-bit equality: (A1 XNOR B1) AND (A0 XNOR B0)
- Seven-segment segment a from BCD with don’t-cares as assumptions
- BCD digit greater than five: four-variable K-map with don’t-cares
- Consensus in practice: AB + A'C + BC simplified
- Absorption in practice: A OR (A AND B) simplified
- De Morgan in practice: NOT (A AND B) rewritten and verified
- The De Morgan slip: NOT (A AND B) vs NOT A AND NOT B
- Distribution in practice: A AND (B OR C) expanded
- XOR built from AND, OR and NOT
- XNOR built from AND, OR and NOT
- Implication as NOT A OR B, verified
- Contrapositive: A → B is equivalent to NOT B → NOT A
- The converse slip: A → B is not B → A
- The inverse slip: A → B is not NOT A → NOT B
- Modus ponens: A → B, A, therefore B (valid)
- Modus tollens: A → B, NOT B, therefore NOT A (valid)
- Affirming the consequent: A → B, B, therefore A (invalid, with counterexample)
- Denying the antecedent: A → B, NOT A, therefore NOT B (invalid)
- Hypothetical syllogism: A → B, B → C, therefore A → C
- Disjunctive syllogism: A OR B, NOT A, therefore B
- Constructive dilemma: (A → B) AND (C → D), A OR C, therefore B OR D
- Inconsistent premises: A and NOT A prove anything (vacuous validity)
- A tautology: A OR NOT A, checked on every row
- A contradiction: A AND NOT A
- Peirce's law: ((A → B) → A) → A is a tautology
- Σm(1, 2, 4, 7): canonical sum of products for three variables
- Four-variable K-map: Σm(0, 2, 5, 7, 8, 10, 13, 15) simplified
- Four-variable product of sums: ΠM(…) simplified from the zeros
- Prime implicants and essential prime implicants on a K-map
- XOR with four NAND gates, verified equivalent
- AND with NOR gates only
- Majority of three with NAND gates only
- Discount eligibility rule: (student OR senior) AND NOT expired AND (member OR coupon)
- Feature flag rule: (beta_user OR internal) AND flag_on AND NOT kill_switch
- Access control rule: owner OR admin OR (editor AND NOT locked)
- Loan eligibility: income_ok AND (credit_ok OR guarantor) AND NOT bankrupt
- Free shipping rule: (over_50 OR prime) AND NOT oversized AND domestic
- Alarm rule: armed AND (door OR window OR motion) AND NOT disarm_code
- Traffic-light safety property: never both greens, checked as “always true”
- Machine interlock: start AND guard_closed AND NOT (estop OR fault)
- Staircase light with three switches: A XOR B XOR C
- Quorum: at least three of four, truth table and K-map
- Leap year rule: (div4 AND NOT div100) OR div400
- Password policy: length_ok AND (has_digit OR has_symbol) AND NOT common
- Exactly one payment option: cash, card or voucher
- Chained conditions: if premium then (discount → approved)
- De Morgan in code: !(a && b) refactored to !a || !b, verified
- A nullable flag rule in SQL: why (a OR NOT a) is not always true
Related
Tools
- Truth table generator: any Boolean expression, up to 12 variables, with intermediate columns
- Boolean algebra calculator and simplifier with steps (every law named and verified)
- Karnaugh map solver: 2 to 4 variables, groups, SOP and POS, don’t-cares from assumptions
Worked examples
Generated from the Tangibool engine, semantics tangibool-semantics/1, engine 1.0.0, build . Every table, map, simplification and verdict on these pages was computed by the Tangibool engine when the site was built, and every claim was re-checked by the same exact checker the app uses. Limits are stated where they apply. © 2026 Foreranger LTD.