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Question 5984: Inclusive Disjunction versus Exactly-One Truth

On this page
  1. Original question and givens
  2. Definition and scope
  3. Step-by-step derivation
  4. Reverse checks
  5. Result
  6. This question review and source binding

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R0010-05984 · q-20261009-advance-20261013-r05-logical-connectives-0084

Original question and givens

Complete classical eight-row tables have indices 0,…,7, with 1=True and 0=False: A=[1, 1, 0, 0, 0, 0, 0, 0], B=[0, 1, 0, 0, 0, 0, 0, 0]. Give (inclusive A∨B truth table, exactly-one-True XOR truth table), distinguishing the two.

  • G001 — Complete classical eight-row tables have indices 0,…,7, with 1=True and 0=False: A=[1, 1, 0, 0, 0, 0, 0, 0], B=[0, 1, 0, 0, 0, 0, 0, 0]. Give (inclusive A∨B truth table, exactly-one-True XOR truth table), distinguishing the two.

Definition and scope

Use the original question domain, assumptions and cited MKO definitions. This item has second-context mathematical and bilingual review, with the model, answer and every arithmetic step independently recomputed from the original parameters; the source question remains a candidate.

Step-by-step derivation

S001 · implies

Before

given8 rows A=[1, 1, 0, 0, 0, 0, 0, 0];B=[0, 1, 0, 0, 0, 0, 0, 0]

After

classical Boolean values;read all supplied rows in order0,...,7

Conditions: none

Interpret1 as True and0 as False and read all eight supplied rows.Do not assume arbitrary eight rows exhaust all valuations or prove every inference rule.Material implication is not causation.

Step identifiers and dependencies

Rule: classical_eight_row_connective_scope · Uses: G001

S002 · implies

Before

row=0;A=True;B=False

After

inclusive A∨B=True;exactly-one XOR=True

Conditions: none

Inclusive disjunction permits both inputs true;exclusive disjunction requires exactly one.They differ when both inputs are true.

Step identifiers and dependencies

Rule: inclusive_or_vs_xor_row · Uses: G001, S001

S003 · implies

Before

row=1;A=True;B=True

After

inclusive A∨B=True;exactly-one XOR=False

Conditions: none

Inclusive disjunction permits both inputs true;exclusive disjunction requires exactly one.They differ when both inputs are true.

Step identifiers and dependencies

Rule: inclusive_or_vs_xor_row · Uses: G001, S002

S004 · implies

Before

row=2;A=False;B=False

After

inclusive A∨B=False;exactly-one XOR=False

Conditions: none

Inclusive disjunction permits both inputs true;exclusive disjunction requires exactly one.They differ when both inputs are true.

Step identifiers and dependencies

Rule: inclusive_or_vs_xor_row · Uses: G001, S003

S005 · implies

Before

row=3;A=False;B=False

After

inclusive A∨B=False;exactly-one XOR=False

Conditions: none

Inclusive disjunction permits both inputs true;exclusive disjunction requires exactly one.They differ when both inputs are true.

Step identifiers and dependencies

Rule: inclusive_or_vs_xor_row · Uses: G001, S004

S006 · implies

Before

row=4;A=False;B=False

After

inclusive A∨B=False;exactly-one XOR=False

Conditions: none

Inclusive disjunction permits both inputs true;exclusive disjunction requires exactly one.They differ when both inputs are true.

Step identifiers and dependencies

Rule: inclusive_or_vs_xor_row · Uses: G001, S005

S007 · implies

Before

row=5;A=False;B=False

After

inclusive A∨B=False;exactly-one XOR=False

Conditions: none

Inclusive disjunction permits both inputs true;exclusive disjunction requires exactly one.They differ when both inputs are true.

Step identifiers and dependencies

Rule: inclusive_or_vs_xor_row · Uses: G001, S006

S008 · implies

Before

row=6;A=False;B=False

After

inclusive A∨B=False;exactly-one XOR=False

Conditions: none

Inclusive disjunction permits both inputs true;exclusive disjunction requires exactly one.They differ when both inputs are true.

Step identifiers and dependencies

Rule: inclusive_or_vs_xor_row · Uses: G001, S007

S009 · implies

Before

row=7;A=False;B=False

After

inclusive A∨B=False;exactly-one XOR=False

Conditions: none

Inclusive disjunction permits both inputs true;exclusive disjunction requires exactly one.They differ when both inputs are true.

Step identifiers and dependencies

Rule: inclusive_or_vs_xor_row · Uses: G001, S008

S010 · implies

Before

([True, True, False, False, False, False, False, False], [True, False, False, False, False, False, False, False])

After

complete ordered results of the two requested connectives

Conditions: none

Each output column retains all eight original rows.This checks the given classical table rowwise,not a general formal proof inferred from matching values or table labels.

Step identifiers and dependencies

Rule: complete_given_connective_rows · Uses: G001, S009

Reverse checks

  • S010 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S009 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S008 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S007 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S006 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S005 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S004 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S003 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S002 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S001 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.

Result

([True, True, False, False, False, False, False, False], [True, False, False, False, False, False, False, False])

The full substitutions and premise checks give the original answer; source identity,format and status are unchanged.

This question review and source binding

Review date: 2026-10-26

Original questions, answers, step expressions, numeric ASTs and historical records are retained. Source candidate labels remain historical and are not overwritten as formal proof.

Source hashes and review receipts
  • input commit: 0f7be61e2c22317a4075561dc822c87d881a50db
  • original question SHA-256: a46c4ce7352b699ccef7d224cf2ad0b5b3a3e6777957163af418bb48fbec0696
  • candidate record SHA-256: ccdb0a9d9523cd53ea32c21d3b6076440fc67a40bbc5d70d736875cc3ac3b516
  • semantic review receipt SHA-256: 0328166a2e335aac7a5fb34556e68c80ba7d16612e822b3999c8ac348a9ca06f
  • question-first witness SHA-256: 0da18e4e6cf71ece5ec5f00dd0cd161217b6d251672bc0b46d79fe7e6253f5fc
  • Lean: NOT_RUN · human review: NOT_PERFORMED
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