From 6fb1b1c8ea337f4ca20b3c039d755a4378bd01ab Mon Sep 17 00:00:00 2001 From: thomasabishop Date: Sat, 17 Dec 2022 16:30:05 +0000 Subject: [PATCH] Autosave: 2022-12-17 16:30:05 --- Logic/Propositional_logic/Boolean_functions.md | 15 +++++++++++++-- .../Corresponding_material_and_biconditional.md | 0 .../DeMorgan's_Laws.md | 0 .../Law_of_Non-Contradiction.md | 0 .../Law_of_the_Excluded_Middle.md | 0 .../Theorems_and_empty_sets.md | 0 6 files changed, 13 insertions(+), 2 deletions(-) rename Logic/{Theorems => _Laws_and_theorems.md}/Corresponding_material_and_biconditional.md (100%) rename Logic/{Theorems => _Laws_and_theorems.md}/DeMorgan's_Laws.md (100%) rename Logic/{Theorems => _Laws_and_theorems.md}/Law_of_Non-Contradiction.md (100%) rename Logic/{Theorems => _Laws_and_theorems.md}/Law_of_the_Excluded_Middle.md (100%) rename Logic/{Theorems => _Laws_and_theorems.md}/Theorems_and_empty_sets.md (100%) diff --git a/Logic/Propositional_logic/Boolean_functions.md b/Logic/Propositional_logic/Boolean_functions.md index ee7d205..1a77adf 100644 --- a/Logic/Propositional_logic/Boolean_functions.md +++ b/Logic/Propositional_logic/Boolean_functions.md @@ -15,7 +15,18 @@ $$ Here is a work through where $f(1, 0, 1)$: - The first disjunction : $ \lnot(x) \land z $ is false because $x$ is 1 and $z$ is 1 -- The second disjunction: $x \land y$ is false because $x$ is 1 and $y$ is 0 +- The second disjunction: $x \land y$ is true because $x$ is 1 and $y$ is 1 - The overall function returns true because the main connective is disjunction and one of the disjuncts (the second) evaluates to 1. Thus the output is 1. -We can compute all possible outputs of the function by constructing a truth-table with each possible variable as the truth conditions and the output of the function as the truth value: +We can compute all possible outputs of the function by constructing a [truth-table](/Logic/Propositional_logic/Truth-tables.md) with each possible variable as the truth conditions and the output of the function as the truth value: + +| $x$ | $y$ | $z$ | $f(x,y,z) = (x \land y) \lor (\lnot(x) \land z )$ | +| --- | --- | --- | ------------------------------------------------- | +| 0 | 0 | 0 | 0 | +| 0 | 0 | 1 | 1 | +| 0 | 1 | 0 | 0 | +| 0 | 1 | 1 | 1 | +| 1 | 0 | 0 | 0 | +| 1 | 0 | 1 | 0 | +| 1 | 1 | 0 | 1 | +| 1 | 1 | 1 | 1 | diff --git a/Logic/Theorems/Corresponding_material_and_biconditional.md b/Logic/_Laws_and_theorems.md/Corresponding_material_and_biconditional.md similarity index 100% rename from Logic/Theorems/Corresponding_material_and_biconditional.md rename to Logic/_Laws_and_theorems.md/Corresponding_material_and_biconditional.md diff --git a/Logic/Theorems/DeMorgan's_Laws.md b/Logic/_Laws_and_theorems.md/DeMorgan's_Laws.md similarity index 100% rename from Logic/Theorems/DeMorgan's_Laws.md rename to Logic/_Laws_and_theorems.md/DeMorgan's_Laws.md diff --git a/Logic/Theorems/Law_of_Non-Contradiction.md b/Logic/_Laws_and_theorems.md/Law_of_Non-Contradiction.md similarity index 100% rename from Logic/Theorems/Law_of_Non-Contradiction.md rename to Logic/_Laws_and_theorems.md/Law_of_Non-Contradiction.md diff --git a/Logic/Theorems/Law_of_the_Excluded_Middle.md b/Logic/_Laws_and_theorems.md/Law_of_the_Excluded_Middle.md similarity index 100% rename from Logic/Theorems/Law_of_the_Excluded_Middle.md rename to Logic/_Laws_and_theorems.md/Law_of_the_Excluded_Middle.md diff --git a/Logic/Theorems/Theorems_and_empty_sets.md b/Logic/_Laws_and_theorems.md/Theorems_and_empty_sets.md similarity index 100% rename from Logic/Theorems/Theorems_and_empty_sets.md rename to Logic/_Laws_and_theorems.md/Theorems_and_empty_sets.md