ECE 2020 Digital Design

Last time

• Last time
  • We talked about Boolean algebras
  • We introduced three operators: NOT, AND, OR
• To be effectively prepared for today you should have:
  • Read textbook Sections 1.2, 1.4, 3.0-3.1 (we have not covered everything yet)
  • Read your notes and reviewed the examples
  • Read the syllabus
• Today
  • We will show why NOR, AND, OR are special
  • We will learn how to manipulate boolean functions
  • We will introduce standard forms of functions as "products of sum" and "sum of products"
  • We probably won't have time to introduce devices that implement boolean functions
• Be ready!
  • You will be asked to doodle a few things on a writing support
  • We will take a quizz on canvas to see if my attendance taking system works

Possible Boolean Functions

• There are \(2^4=16\) possible boolean functions of two variables \(F(A,B)\)

• We can list them all exhaustively

• We can also express them as a combination of NOT, OR, and AND

Operator precedence

• Operator precedence: there is a rule of precedence (similar to our usual rules)
  • parentheses
  • NOT
  • AND
  • OR
• Example: \[ F = (A\overline{(C+\overline{B}\cdot D)}+\overline{B\cdot \overline{C}})\overline{E} \]

Evaluation of functions

• Example 1: evaluate \(F=C+\overline{C}\cdot B+B\cdot \cdot \overline{A}\) with \(A=1\), \(B=0\), \(C=1\)

• Example 2: evaluate \(F=D(B\cdot \overline{C}\cdot A+\overline{(A\cdot \overline{B}+C)}+C)\) with \(A=0, B=0, C=1, D=1\)

Axiomatic construction of Boolean algebra

• Objective: define rules to manipulate functions and variables without directly evaluating

• Axioms:
  • If \(X\neq 1\) then \(X=0\) and vice versa
  • Tables of NOT, AND, OR
• We can now prove a lot of "theorems" that make the manipulation of boolean functions easier

Simplifications of functions

• Example 1: \(F = B\cdot C+B\cdot \overline{C} + B\cdot A\)

• Example 2: \(F=A + \overline{A}\cdot B+\overline{A}\cdot\overline{B}\cdot C + \overline{A}\cdot\overline{B}\cdot\overline{C}\cdot D+\overline{A}\cdot\overline{B}\cdot\overline{C}\cdot\overline{D}\cdot E\)

• Example 3: Show that \(\overline{A(\overline{B}\cdot \overline{C}+B\cdot C)}=\overline{A}+(B+C)(\overline{B}+\overline{C})\)

SOP and POS forms

• We know how to create a truth table from a function

• How do we create a function from a truth table
  • There is no unique way…
  • but there are standard ways
• We will introduce sum of products and product of sums

Until next time

• To be effectively prepared for Monday August 25, 2025, you should:
  • Read textbook Sections 1.2, 1.4, 3.0-3.1 (we have covered most things)
  • Read your notes and review the examples
• Next time
  • We will talk about Karnaugh maps
  • We will introduce devices that implement boolean functions
• What to expect later this week
  • More detailed tentative schedule
  • Office hours location and time
  • First homework!