Lecture 5 - Boolean Logic & Circuits

Boolean Algebra and Digital Design

Introduction to Boolean Algebra

• Boolean algebra is a system of rules and operations used with variables that can only have two values: $0$ (false) or $1$ (true).

• It forms the mathematical foundation for digital electronic circuits and computer logic.

Boolean Operations

• Boolean Multiplication (AND operation): Denoted by $\cdot$ (or implied by juxtaposition), analogous to logical AND.

  • $0 \cdot 0 = 0$

  • $0 \cdot 1 = 0$

  • $1 \cdot 0 = 0$

  • $1 \cdot 1 = 1$

• Boolean Addition (OR operation): Denoted by $+$, analogous to logical OR.

  • $0 + 0 = 0$

  • $0 + 1 = 1$

  • $1 + 0 = 1$

  • $1 + 1 = 1$

• Boolean Complement (NOT operation): Denoted by a bar symbol ($\overline{x}$) or a prime symbol ($x'$), analogous to logical NOT.

  • $\overline{0} = 1$

  • $\overline{1} = 0$

Examples of Boolean Expression Evaluation

• Example 1: Evaluate the expression derived from the steps $(x \cdot y \cdot z)' + (w \cdot z)' + x$ with $x=1, y=1, w=0, z=0$

  • Substitute values: $(1 \cdot 1 \cdot 0)' + (0 \cdot 0)' + 1$

  • Perform inner multiplications: $(0)' + (0)' + 1$

  • Perform complements: $1 + 1 + 1$

  • Perform additions: $1$

• Example 2: Evaluate $x(y+z)$ with $x=1, y=0, z=1$

  • Substitute values: $1(0+1)$

  • Perform inner addition: $1(1)$

  • Perform multiplication: $1$

• Example 3: Evaluate $xy+z$ with $x=1, y=0, z=1$

  • Substitute values: $1 \cdot 0 + 1$

  • Perform multiplication: $0 + 1$

  • Perform addition: $1$

Boolean Functions

• A Boolean function maps one or more Boolean input variables (from the set ${0, 1}$) to a single Boolean output value (either $0$ or $1$).

• Boolean functions can be defined in two ways:

  • Using a Boolean expression: For example, $f(x, y, z) = xy + yz$.

  • Using an input/output table: Similar to a truth table, it lists the output value for every possible combination of input values.

Constructing an Input/Output Table from a Boolean Expression

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• Example: Construct the input/output table for $f(x, y, z) = xy + yz$
  
  

  
  

  $x$	$y$	$z$	$xy$	$yz$	$f(x,y,z) = xy + yz$
  $0$	$0$	$0$	$0$	$0$	$0$
  $0$	$0$	$1$	$0$	$0$	$0$
  $0$	$1$	$0$	$0$	$0$	$0$
  $0$	$1$	$1$	$0$	$1$	$1$
  $1$	$0$	$0$	$0$	$0$	$0$
  $1$	$0$	$1$	$0$	$0$	$0$
  $1$	$1$	$0$	$1$	$0$	$1$
  $1$	$1$	$1$	$1$	$1$	$1$
  Constructing a Boolean Expression from an Input/Output Table

  • A Boolean expression can be created by summing the minterms of all rows where the function's output is $1$.

    • Literal: A variable appearing in an expression in its true form (e.g., $x$) or complemented form (e.g., $x'$ or $\overline{x}$).

    • Minterm: A product term consisting of exactly one literal for every input variable of the function. For three variables $x, y, z$, a minterm would be a product like $x'yz$ or $xyz'$.

  • Example: Find the Boolean expression for the function defined by the table (from page 5 for $f(x,y,z) = xy+yz$):

    $x$	$y$	$z$	$f(x,y,z)$
    $0$	$0$	$0$	$0$
    $0$	$0$	$1$	$0$
    $0$	$1$	$0$	$0$
    $0$	$1$	$1$	$1$
    $1$	$0$	$0$	$0$
    $1$	$0$	$1$	$0$
    $1$	$1$	$0$	$1$
    $1$	$1$	$1$	$1$

    • Identify rows where $f(x,y,z) = 1$:

      • Row $4$: $x=0, y=1, z=1 \rightarrow$ Minterm: $x'yz$

      • Row $7$: $x=1, y=1, z=0 \rightarrow$ Minterm: $xyz'$

      • Row $8$: $x=1, y=1, z=1 \rightarrow$ Minterm: $xyz$

    • Sum these minterms to get the expression:
      $f(x, y, z) = x'yz + xyz' + xyz$

  Laws of Boolean Algebra

  • These laws are fundamental for manipulating and simplifying Boolean expressions.

  Rule	Application 1 (OR form)	Application 2 (AND form)
  Idempotent Laws	$x + x = x$	$x \cdot x = x$
  Associative Laws	$(x + y) + z = x + (y + z)$	$(x \cdot y) \cdot z = x \cdot (y \cdot z)$
  Commutative Laws	$x + y = y + x$	$x \cdot y = y \cdot x$
  Distributive Laws	$x + (y \cdot z) = (x + y) \cdot (x + z)$	$x \cdot (y + z) = (x \cdot y) + (x \cdot z)$
  Identity Laws	$x + 0 = x$	$x \cdot 1 = x$
  Domination Laws	$x + 1 = 1$	$x \cdot 0 = 0$
  Double Complement Law	$(x')' = x$
  Complement Laws	$x + x' = 1$	$x \cdot x' = 0$
  De Morgan's Laws	$(x + y)' = x' \cdot y'$	$(x \cdot y)' = x' + y'$
  Absorption Laws	$x + (x \cdot y) = x$	$x \cdot (x + y) = x$

  Example of Boolean Expression Simplification

  • Simplify: $x + x'y$

    1. $x + x'y = (x + x')(x + y)$ (Distributive Law: $A+BC = (A+B)(A+C)$; here $A=x, B=x', C=y$)

    2. $(x + x')(x + y) = 1 \cdot (x + y)$ (Complement Law: $x + x' = 1$)

    3. $1 \cdot (x + y) = x + y$ (Identity Law: $1 \cdot A = A$)

  • Therefore, $x + x'y = x + y$

  Logic Gates and Combinational Circuits

  • Gates: Electrical devices that implement Boolean functions. They are the building blocks of digital circuits.

  • Combinational Circuits: Circuits built by combining inverters, OR gates, and AND gates. Their output depends solely on the current inputs.

  Common Logic Gate Symbols

  • AND Gate: Represents Boolean multiplication ($xy$).

    • Symbol: D-shaped, with two inputs ($x, y$) and one output ($xy$).

  • OR Gate: Represents Boolean addition ($x+y$).

    • Symbol: Curved front, with two inputs ($x, y$) and one output ($x+y$).

  • Inverter (NOT Gate): Represents Boolean complement ($x'$).

    • Symbol: Triangle with a small circle (bubble) at the output, one input ($x$) and one output ($x'$).

  Example of a Combinational Circuit and its Boolean Function

  • Consider a circuit with inputs $x, y, z$:

    1. An inverter acts on $x$, producing $x'$.

    2. An OR gate takes $x'$ and $y$ as inputs, producing $x'+y$.

    3. An AND gate takes $(x'+y)$ and $z$ as inputs, producing $(x'+y)z$.

  • Thus, the circuit corresponds to the Boolean function $f(x, y, z) = (x'+y)z$.

  Digital Design Process

  • Designing digital circuits systematically involves several steps:

    1. English Description: Clearly define the desired circuit behavior in natural language.

    2. Input/Output Table: Translate the English description into a formal table showing outputs for all possible input combinations.

    3. Boolean Expression: Derive a Boolean expression from the input/output table (typically as a sum of minterms).

    4. Minimization: Simplify the Boolean expression using the laws of Boolean algebra (or other minimization techniques) to reduce the complexity of the final circuit.

    5. Digital Circuit: Implement the minimized Boolean expression using logic gates.

  Digital Design Example: Comparing Two One-Bit Numbers

  • English Description: Design a circuit that takes two one-bit binary numbers ($x$ and $y$) as inputs and produces a $true$ output (logical $1$) when $x$ is greater than or equal to $y$ ($x \ge y$), and $false$ (logical $0$) otherwise.

  • Input/Output Table:

    $x$	$y$	$f(x,y)$ ($x \ge y$)
    $0$	$0$	$1$ ($0 \ge 0$ is true)
    $0$	$1$	$0$ ($0 \ge 1$ is false)
    $1$	$0$	$1$ ($1 \ge 0$ is true)
    $1$	$1$	$1$ ($1 \ge 1$ is true)

  • Boolean Expression (Sum of Minterms):

    • The rows where $f(x,y)=1$ are:

      • $x=0, y=0 \rightarrow x'y'$

      • $x=1, y=0 \rightarrow xy'$

      • $x=1, y=1 \rightarrow xy$

    • Summing these minterms:
      $f(x,y) = x'y' + xy' + xy$

  • Minimization:

    • $f(x,y) = x'y' + xy' + xy$

    • $= (x'y' + xy') + xy$ (Associative Law)

    • $= (x' + x)y' + xy$ (Distributive Law: factoring out $y'$)

    • $= (1)y' + xy$ (Complement Law: $x' + x = 1$)

    • $= y' + xy$ (Identity Law: $1 \cdot A = A$)

    • $= (y' + x)(y' + y)$ (Distributive Law: $A + BC = (A+B)(A+C)$; here $A=y', B=x, C=y$)

    • $= (y' + x)(1)$ (Complement Law: $y' + y = 1$)

    • $= y' + x$ (Identity Law: $A \cdot 1 = A$)

    • Rearranging: $f(x,y) = x + y'$

  • Digital Circuit:

    • The minimized expression is $f(x,y) = x + y'$. This can be built using one inverter and one OR gate:

      1. An inverter takes $y$ as input to produce $y'$.

      2. An OR gate takes $x$ and $y'$ as inputs, producing the final output $x + y'$.