Discrete Mathematics Lecture Notes: Predicates and Quantifiers

Introduction

• Welcome back to Ramadin Maths Academy.
• Apologies for the delay in posting videos due to Wi-Fi issues.
• Today's lecture focuses on predicates in discrete mathematics.

Key Concepts

Overview of Logic Types

• Two types of logics:
  1. Propositional Logic
  2. Predicate Logic

What is a Predicate?

• Defined in the context of a statement.
• Example:
  • If the statement is "x is greater than 3":
    • x is the variable (subject).
    • Greater than 3 is the predicate.

Notation

• Denote predicate as p(x) where x is the variable.
• Example:
  • p(x) = "x is greater than 3"

Truth Values of Predicates

• Predicates can either be true or false.
• Example of truth values:
  • p(4):
    • True since 4 > 3.
  • p(2):
    • False since 2 > 3 is incorrect.

Further Examples

• Consider another predicate:
  • q(x, y) = "x is equal to y + 3"
  • Determine truth values for different pairs:
    • q(1, 2): False (1 ≠ 5)
    • q(3, 0): True (3 = 0 + 3)

Compound Statements

• Combine two or more predicates using connectives:
  • Conjunction (and)
  • Disjunction (or)
  • Implication (if... then)

Example of Compound Statement

• Statement: "Rama is a teacher and her teaching is good."
  • Identify subjects and predicates:
    • Subject: Rama
    • Predicate: Teacher
    • Combine using:
      • p(r) for "Rama is a teacher"
      • g(t) for "Her teaching is good"
      • Formulate as: p(r) AND g(t)

Quantifiers

Definition of Quantifiers

• Quantifiers express the quantity of elements in a predicate.
  • Universal Quantifier: "For all" or "For every" (denoted as ∀)
  • Existential Quantifier: "There exists" (denoted as ∃)

Universal Quantifier

• Definition: For all values of x in a domain.
• Example:
  • Predicate: p(x) = "x + 1 > x"
  • Truth value: True for all real numbers.

Existential Quantifier

• Definition: There exists at least one element.
• Example:
  • Predicate: p(x) = "x = x + 3"
  • Truth value: False for all real numbers (no solution exists).

Conclusion

• Review of predicates, their truth values, compound statements, and quantifiers.
• Important concepts for understanding discrete mathematics.