eolas/Logic/Propositional_logic/Boolean_algebra.md

---
categories:
  - Logic
tags: [propositional-logic, algebra]
---

# Boolean algebra

Many of the laws that obtain in the mathematical realm of algebra also obtain for Boolean expressions.

## The Commutative Law

$$
    x \land y = y \land x \\
$$

$$

    x \lor y = y \lor x
$$

Compare the [Commutative Law](/Mathematics/Prealgebra/Whole_numbers.md#the-commutative-property) in the context of arithmetic.

## The Associative Law

$$
    x   \land (y \land z) = (x \land y) \land z
$$

$$
    x   \lor (y \lor z) = (x \lor y) \lor z
$$

Compare the [Associative Law](/Mathematics/Prealgebra/Whole_numbers.md#the-associative-property) in the context of arithmetic.

## The Distributive Law

$$
    x \land (y \lor z) = (x \land y) \lor (x \land z)
$$

$$
    x \lor (y \land z) = (x \lor y) \land (x \lor z)
$$

Compare for instance how this applies in the case of [multiplication](/Mathematics/Prealgebra/Distributivity.md):

$$
    a \cdot (b + c) = a \cdot b + a \cdot c
$$

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