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Lesson 9.6 • Competency 9.4Complete Lesson
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Digital Logic Gates & Boolean Algebra

Explore binary logic levels (0 and 1), basic gates (NOT, AND, OR), universal gates (NAND, NOR), exclusive gates (XOR, XNOR), truth tables, Boolean theorems, De Morgan’s Laws, and sensor-based switching circuits.

Marking a lesson complete represents finishing your study of the topic; it does not denote tested mastery.

Learning Objectives

  • Define binary logic levels and construct truth tables for basic gates (NOT, AND, OR), universal gates (NAND, NOR), and XOR/XNOR.
  • Explain why NAND and NOR are universal gates and demonstrate the synthesis of NOT, AND, and OR using NAND/NOR exclusively.
  • Simplify complex digital logic expressions using Boolean theorems and De Morgan’s Laws: NOT(A+B) = NOT(A)·NOT(B).

Core Theory & Concepts

Digital electronics abstracts continuous physical voltages into discrete binary logic states: High (1, typically +5 V) and Low (0, typically 0 V). Combining interconnected logic gates enables the design of digital calculators, computer processors, and automated industrial control systems.

Universal Logic Synthesis

Any Boolean function can be implemented using exclusively NAND gates (or exclusively NOR gates). Connecting both inputs of a NAND together produces an inverter NOT.

\overline{A \cdot A} = \overline{A}, \quad \overline{\overline{A \cdot B}} = A \cdot B

De Morgan’s Laws

De Morgan’s theorems allow conversions between AND and OR operations: The complement of an OR is the AND of complements, and vice versa.

\overline{A + B} = \overline{A} \cdot \overline{B}, \quad \overline{A \cdot B} = \overline{A} + \overline{B}

Common Misconceptions

Misconception

An XOR gate produces output 1 when all its inputs are 1.

Physical Clarification:In a 2-input XOR gate, output is 1 only when inputs differ. When both inputs are 1 (or both 0), the output is 0.

Syllabus Topic Outline

Competency 9.4
  • 1Binary logic levels: High (1) and Low (0) voltage thresholds
  • 2Basic gates: Inverter (NOT), Conjunction (AND), Disjunction (OR)
  • 3Universal gates: NAND and NOR topologies and circuit synthesis
  • 4Exclusive gates: XOR and XNOR applications in adders and parity checking
  • 5Boolean algebra theorems and De Morgan’s Laws simplification
  • 6Sensor-driven digital switching: LDR, thermistor, and transistor interface

Practice Problems

1
Competency 9.4 • foundation

Which logic gate produces an output of logic 0 ONLY when both of its inputs are logic 1?