From 1cb58e93aab60f1b8940f7b04196ea5afb3640d0 Mon Sep 17 00:00:00 2001 From: thomasabishop Date: Tue, 20 Dec 2022 07:00:05 +0000 Subject: [PATCH] Autosave: 2022-12-20 07:00:05 --- Logic/Propositional_logic/Boolean_function_synthesis.md | 8 ++++---- 1 file changed, 4 insertions(+), 4 deletions(-) diff --git a/Logic/Propositional_logic/Boolean_function_synthesis.md b/Logic/Propositional_logic/Boolean_function_synthesis.md index dab9bf3..e6c5798 100644 --- a/Logic/Propositional_logic/Boolean_function_synthesis.md +++ b/Logic/Propositional_logic/Boolean_function_synthesis.md @@ -7,9 +7,9 @@ tags: [logic, propositional-logic, nand-to-tetris] # Boolean function synthesis -When we looked at [boolean functions](/Logic/Propositional_logic/Boolean_functions.md) we were working in a particular direction: from a function to a truth table. When we do Boolean function synthesis we work in the opposite direction: from a function to a truth table. +When we looked at [boolean functions](/Logic/Propositional_logic/Boolean_functions.md) we were working in a particular direction: from a function to a truth table. When we do Boolean function synthesis we work in the opposite direction: from a truth table to a function. -This is an important skill that we will use when constructing [logic circuits](/Electronics_and_Hardware/Digital_circuits/Digital_circuits.md). We will go from truth conditions (i.e. what we want the circuit to do and when we want it to do it) to a function expression which is then reduced and implemented with [logic gates](/Electronics_and_Hardware/Digital_circuits/Logic_gates.md). +This is an important skill that we will use when constructing [logic circuits](/Electronics_and_Hardware/Digital_circuits/Digital_circuits.md). We will go from truth conditions (i.e. what we want the circuit to do and when we want it to do it) to a function expression which is then reduced to its simplest form and implemented with [logic gates](/Electronics_and_Hardware/Digital_circuits/Logic_gates.md). ## The process @@ -52,7 +52,7 @@ For each line we construct a Boolean expression that would result in the value i | 3 | 0 | 1 | 0 | $ \lnot(x) \land y \land \lnot(z) $ | | 5 | 1 | 0 | 0 | $ x \land \lnot(y) \land \lnot(z) $ | -We can now join each expression to create a complex expression that covers the entire truth table. Since 1 will be output for any one of these sub-expressions we can just join them up with OR: +We can now join each expression to create a complex expression that covers the entire truth table using OR: $$ (\lnot(x) \land \lnot (y) \land \lnot(z)) \lor (\lnot(x) \land y \land \lnot(z)) \lor (x \land \lnot(y) \land \lnot(z)) @@ -69,7 +69,7 @@ $$ (\lnot(x) \land \lnot(z)) \lor (\lnot(y) \land \lnot(z)) $$ -Notice that $\lnot(z)$ is repeated so we can remove the repetition: +Notice that $\lnot(z)$ is repeated so we can remove the repetition through [idempotence](/Logic/Propositional_logic/Boolean_algebra.md#idempotent-law): $$ \lnot z \land (\lnot(x) \lor \lnot(y))