Lecture 8 - Boolean Algebras and their Representations

Introduction to Boolean Algebras

  • Fundamental to understanding digital computing and logic.

  • Consists of bits governed by Boolean algebra.

Basic Concepts in Digital Computing

  • Bits: Basic units of information, can be in state 0 or 1.

  • Boolean Operations: Include operations such as NOT, AND, OR which allow manipulation of bits.

  • Every piece of information can be encoded in sequences of bits.

Transition to Quantum Computing

  • Basic unit of information in quantum computing is the qubit.

  • Quantum computing utilizes complex linear algebra for processing information.

What is Boolean Algebra?

  • Boolean Algebras are algebraic structures consisting of:

    • A set of at least two distinct elements, denoted as B.

    • A bottom element ⊥ (representing false).

    • A top element ⊥ (representing true).

    • Two binary operations: ∧ (meet) and ∨ (join).

    • One unary operation, denoted as ¬ (complement).

Axioms of Boolean Algebra

  • Identity elements:

    • For any element x in B,
      x∧⊥=xx ∧ ⊥ = x
      x∨⊥=⊥x ∨ ⊥ = ⊥

  • Complement laws:

    • x∧¬x=⊥x ∧ ¬x = ⊥

    • x∨¬x=⊤x ∨ ¬x = ⊤

  • Commutativity:

    • x∧y=y∧xx ∧ y = y ∧ x

    • x∨y=y∨xx ∨ y = y ∨ x

  • Associativity:

    • x∧(y∧z)=(x∧y)∧zx ∧ (y ∧ z) = (x ∧ y) ∧ z

    • x∨(y∨z)=(x∨y)∨zx ∨ (y ∨ z) = (x ∨ y) ∨ z

  • Distributivity:

    • x∧(y∨z)=(x∧y)∨(x∧z)x ∧ (y ∨ z) = (x ∧ y) ∨ (x ∧ z)

    • x∨(y∧z)=(x∨y)∧(x∨z)x ∨ (y ∧ z) = (x ∨ y) ∧ (x ∨ z)

Examples of Boolean Algebras

  1. Boolean Logic:

    • Set B = {F, T} (F = false, T = true)

    • Operations:

      • ⊥≡F⊥ ≡ F

      • ⊤≡T⊤ ≡ T

      • ∧≡AND∧ ≡ \text{AND}

      • ∨≡OR∨ ≡ \text{OR}

      • ¬≡NOT¬ ≡ \text{NOT}

  2. Power Sets:

    • For any non-empty set X, the power set (set of all subsets) forms a Boolean algebra.

    • Example: If n = |X|, then the Boolean algebra B = 2X{2^X} with:

      • ⊥≡∅⊥ ≡ ∅ (empty set)

      • ⊤≡X⊤ ≡ X

      • ∧≡∩∧ ≡ ∩ (intersection)

      • ∨≡∪∨ ≡ ∪ (union)

      • Complement represented as X\YX \backslash Y

Properties of Boolean Algebras

  • A Boolean algebra is a complemented distributive lattice.

  • Each element has a unique complement.

  • The structure satisfies absorbtion laws: x∧(x∨y)=xx ∧ (x ∨ y) = x and x∨(x∧y)=xx ∨ (x ∧ y) = x.

Representation of Boolean Logic

  • 1st Representation:

    • F ≡ 0, T ≡ 1

    • x∧y≡min(x,y)x ∧ y ≡ \text{min}(x, y)

    • x∨y≡max(x,y)x ∨ y ≡ \text{max}(x, y).

    • Notably utilized in fuzzy logic.

  • 2nd Representation (Boolean Rings):

    • A Boolean ring R with operations + and ·;

    • Properties that create equivalences to logical operations including AND and OR.

Operations in Boolean Logic

  • Basic logical operations (truth tables):

    • NOT: ¬x¬x

    • AND: x∧yx ∧ y

    • OR: x∨yx ∨ y

    • XOR: x⊕yx ⊕ y with x⊕y=(x∧¬y)∨(¬x∧y)x ⊕ y = (x ∧ ¬y) ∨ (¬x ∧ y)

    • NAND: the negation of AND.

Arithmetic and Boolean Logic

  • Boolean logic serves as the foundation for arithmetic operations in digital circuits.

  • Examples include half adders and full adders, which map inputs to sum and carry outputs based on logical operations.

Conclusion

  • Understanding Boolean algebras and their representations is essential for grasping digital computing and quantum information processing standards.