52 lines
1.1 KiB
Markdown
52 lines
1.1 KiB
Markdown
---
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categories:
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- Logic
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tags: [propositional-logic, algebra]
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---
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# Boolean algebra
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Many of the laws that obtain in the mathematical realm of algebra also obtain for Boolean expressions.
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## The Commutative Law
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$$
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x \land y = y \land x \\
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$$
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$$
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x \lor y = y \lor x
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$$
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Compare the [Commutative Law](/Mathematics/Prealgebra/Whole_numbers.md#the-commutative-property) in the context of arithmetic.
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## The Associative Law
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$$
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x \land (y \land z) = (x \land y) \land z
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$$
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$$
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x \lor (y \lor z) = (x \lor y) \lor z
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$$
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Compare the [Associative Law](/Mathematics/Prealgebra/Whole_numbers.md#the-associative-property) in the context of arithmetic.
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## The Distributive Law
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$$
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x \land (y \lor z) = (x \land y) \lor (x \land z)
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$$
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$$
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x \lor (y \land z) = (x \lor y) \land (x \lor z)
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$$
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Compare for instance how this applies in the case of [multiplication](/Mathematics/Prealgebra/Distributivity.md):
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$$
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a \cdot (b + c) = a \cdot b + a \cdot c
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$$
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In addition we have [DeMorgan's Laws](/Logic/Laws_and_theorems.md/DeMorgan's_Laws.md) which express the relationship that obtains between the negations of conjunctive and disjunctive expressions
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