MTH202 - Discrete Mathematics: Lecture Notes

MTH202 – Discrete Mathematics Lecture Notes

Lecture #5: Valid and Invalid Arguments

• Example of Argument:

  • Statement: An interesting teacher keeps me awake. I stay awake in Discrete Mathematics class. Therefore, my Discrete Mathematics teacher is interesting.
  • Question: Is the above argument valid?

• Definition of Argument:

  • An argument consists of a list of statements called premises (or assumptions or hypotheses) followed by a statement called the conclusion.
  • Structure of Argument:
  • P1 (Premise)
  • P2 (Premise)
  • P3 (Premise)
  • …
  • Pn (Premise)
  • Conclusion (C):
  • Symbolically represented as:
    $ext{P1} \, ext{P2} \, ext{P3} \, … \, ext{Pn} \ herefore C$
  • Note: The symbol $herefore$ is read as "therefore" and is conventionally positioned just before the conclusion.

Valid and Invalid Arguments

• Valid Argument: An argument is valid if the conclusion is true when all the premises are true.
  • Alternatively, it is valid if the conjunction of its premises implies the conclusion:
    $(P1 \land P2 \land P3 \land … \land Pn) <br /> ightarrow C$ is a tautology.
• Invalid Argument: An argument is invalid if the conclusion is false when all the premises are true.
  • Alternatively, it is invalid if the conjunction of its premises does not imply the conclusion.

Example: Valid Argument Form

• Argument Structure:
  • Premises:
  • $p <br /> ightarrow q$
  • $p$
  • Conclusion:
  • $herefore q$
• Solution:
  • Conclusions derived demonstrate validity based on given premises.

Example: Invalid Argument Form

• Argument Structure:
  • Premises:
  • $p <br /> ightarrow q$
  • $q$
  • Conclusion:
  • $herefore p$
• Truth Table Analysis:
  • Combination of truth values lead to invalidity conditions for premises and conclusion.

Lecture #6: Logic Gates and Circuits

Switches in Series and Parallel

• Switches in Series:
  • Open circuit state leads to light bulb OFF.
  • Closed circuit states:
  • Open/Closed pairs yield varying light bulb states (ON/OFF).
• Switches in Parallel:
  • Allows for independent ON states across switches.

Basic Logic Gates

1. NOT-gate (Inverter):

   • Definition: A circuit with one input and one output.
   • Operation:
     • Input = 1, Output = 0
     • Input = 0, Output = 1

2. AND-gate:

   • Definition: A circuit with two input signals and one output signal.
   • Operation:
     • Both inputs = 1, Output = 1
     • Otherwise, Output = 0

3. OR-gate:

   • Definition: A circuit with two input signals and one output signal.
   • Operation:
     • Both inputs = 0, Output = 0
     • Otherwise, Output = 1

Constructing Input/Output Tables

• Combinational Circuit:
  • Represents logic circuits using basic gates.
• Exercise:
  • Determine output for given inputs using AND, OR, NOT gates in a complex circuit.

Finding Boolean Expressions

• Constructing Boolean Expression:
  • Trace through circuits to deduce input/output behavior and formulate corresponding Boolean expressions.
  • Example expression derived from supplementary tables illustrates logical equivalence of circuit outputs.

Exercises in Logic Gates

• Boolean expressions must uphold logical equivalence between varied circuit arrangements.
• Demonstrations of identity and negation laws support validity assessments between expressions:
  • Example expressions analyzed using logical rules (e.g., distributive, identity laws) leading to conclusions on circuit equivalences.

Additional Exercises

• Circuit Construction:
  • Given Boolean expressions, students are tasked with constructing circuits.
  • Inputs/Outputs should follow assessed logical outputs based on designed arrangements.

Summary of Key Concepts:

• Arguments, validity, and the structure of logical reasoning in discrete mathematics.
• Logic gates characterized by behaviors and output configurations contributing to circuit logic.
• Formulation and analysis of Boolean expressions and statements yielding equivalent logical constructs.