Rules of Boolean Algebra

A Boolean Algebra is a mathematical structure consisting of (1) a set together with (2) operations and distinguished elements:

• The set typically contains elements such as \( 0 \) and \( 1 \), which represent false and true.
• The main operations are conjunction $\land$ (

  $\land$

  ), disjunction $\lor$ (

  $\lor$

  ), and negation/inversion $\neg$ (

  $\neg$

  ).
  • The operation \( \land \) represents logical AND, and it returns true only when both operands are true.
  • The operation \( \lor \) represents logical OR, and it returns true when at least one operand is true.
  • Lastly, the operation \( \neg \) represents logical NOT, and it inverts the truth value of an element.

Example: \( 1 \land 0 = 0 \), and \( 1 \lor 0 = 1 \). Also, \( \neg 1 = 0 \), and \( \neg 0 = 1 \). Formally, a Boolean Algebra can be written as \( (B, \lor, \land, \neg, 0, 1) \).

Interesting observation: we can substitute \( \land \) with $\cdot$ (

$\cdot$

) (multiplication) and \( \lor \) with $+$ (addition), and have these operations work almost completely fine as with integer arithmetics: $0 \cdot 0 = 0$, $0 \cdot 1 = 0$, $1 \cdot 0 = 0$, $1 \cdot 1 = 1$, $0 + 0 = 0$, $0 + 1 = 1$, $1 + 0 = 1$, but $1 + 1 = 1$.