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Fundamentals of Discrete Mathematics

Feb 1, 2025

Discrete Mathematics Lecture Notes

Swagat

  • Welcome to Pradeep Giri AkaiGwi channel.
  • Demand for DM lectures was made.

Objective of the Lecture

  • Discuss the syllabus of discrete mathematics.
  • Theory questions are also asked.

Set Theory

  • Definition: A collection of well-defined objects.
    • Example: Natural numbers less than 5 (1, 2, 3, 4).
  • Representation of Sets:
    • Sets are always represented by uppercase letters.
    • Example: If set тАШAтАЩ, it is written in curly brackets.

Ways to Represent a Set

  1. Roster Method:
    • Useful for limited elements.
    • Example: Days of the week (Monday, Tuesday, ...).
  2. Set Builder Form:
    • When the number of elements is more.
    • Example: x | x < 70 (x is a natural number).

Different Types of Numbers

  • Natural Numbers:
    • Represented by тАШNтАЩ.
    • Starts from 1 and goes to infinity.
  • Integers:
    • Represented by тАШZтАЩ.
    • Zero and positive and negative numbers on either side.
  • Rational Numbers:
    • Represented by тАШQтАЩ.
    • In the form p/q, where q тЙа 0.
  • Irrational Numbers:
    • Numbers whose square root does not resolve.
  • Real Numbers:
    • All types of numbers.
  • Complex Numbers:
    • Represented in the form of i.

Types of Sets

  1. Singleton Set:
    • A set with a single element.
  2. Null Set:
    • A set with no elements.
  3. Finite Set:
    • A set with limited elements.
  4. Infinite Set:
    • A set with infinite elements.

Relationships within Sets

  • Subset:
    • If all elements are in one set.
  • Superset:
    • If a set has all the elements of another set.
  • Power Set:
    • Set of all possible subsets.
    • Formula: 2^n, where n is the number of elements in the set.

Special Types of Sets

  • Disjoint Set:
    • Elements of two sets differ from each other.
  • Overlapping Set:
    • At least one common element.
  • Equivalent Set:
    • Both sets have the same number of elements.
  • Equal Set:
    • Both sets have the same elements, order may differ.

Cardinality of Set

  • The number of elements in a set is called its cardinality.
  • Example: Cardinality of {1, 2, 3, 4} is 4.

Next Lecture

  • Discussion on Venn Diagrams in the next lecture.
  • Focus will be on solving important problems.

Conclusion

  • Today's lecture was full of important information.
  • Do not miss the next class.

Full transcript