33 lines
1.5 KiB
Markdown
33 lines
1.5 KiB
Markdown
---
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categories:
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- Logic
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- Computer Architecture
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tags: [logic, propositional-logic, nand-to-tetris]
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---
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# Boolean functions
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An example of a Boolean function:
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$$
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f(x,y,z) = (x \land y) \lor (\lnot(x) \land z )
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$$
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Here is a work through where $f(1, 0, 1)$:
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- The first disjunction : $\lnot(x) \land z$ is false because $x$ is 1 and $z$ is 0
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- The second disjunction: $x \land y$ is false because $x$ is 1 and $y$ is 1
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- The overall function returns false because the main connective is disjunction and both of its disjuncts are false
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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:
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| $x$ | $y$ | $z$ | $f(x,y,z) = (x \land y) \lor (\lnot(x) \land z )$ |
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| --- | --- | --- | ------------------------------------------------- |
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| 0 | 0 | 0 | 0 |
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| 0 | 0 | 1 | 1 |
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| 0 | 1 | 0 | 0 |
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| 0 | 1 | 1 | 1 |
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| 1 | 0 | 0 | 0 |
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| 1 | 0 | 1 | 0 |
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| 1 | 1 | 0 | 1 |
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| 1 | 1 | 1 | 1 |
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