This document discusses predicates and quantifiers in predicate logic. It begins by defining predicate logic as an extension of propositional logic that allows reasoning about whole classes of entities. It then discusses predicates, subjects, and the universal and existential quantifiers. The universal quantifier is defined using the example "All parking spaces at BU are full" while the existential quantifier is defined using the example "There is a parking space at BU that is full." Finally, it discusses some quantifier equivalence laws.

1. 1
PresentationTopic:
Predicates and Quantifiers
Discrete Mathematics

2. Submitted By: Mehedi Hasan Saikat
ID: 162-15-7746
Section: B
Department Of Computer Science & Engineering
Daffodil International University.
SubmittedTo: Md. Sadekur Rahman
Senior Lecturer
Department Of Computer Science & Engineering.
Daffodil International University.

3. Predicate Logic
•Predicate logic is an extension of propositional logic
•It permits concisely reasoning about whole classes of
entities.
•Examples of a class is an integer class, a student in
CSE Dept, etc.

4. Propositional Logic
•Atomic propositions: p, q, r, …
•Boolean operators:      
•Compound propositions: (p  q)  r
•Equivalences: pq ≡ (p  q)
•Proving equivalences using:
• Truth tables.
• Symbolic derivations (Laws).

5. Applications of Predicate Logic
• It is the formal notation for writing perfectly clear, concise, and
unambiguous mathematical definitions, axioms, and theorems for
any branch of mathematics.
• Statements like x > 5 are neither true nor false when the value of
x is not specified.
• Predicate logic can be used to make propositions from such
statements.

6. Subjects and Predicates
• Example “The dog is sleeping”:
• In predicate logic, a predicate is modeled as a function
P(x ) from objects to propositions.
• P(x) = “x is sleeping” (where x is any object).

7. The Universal Quantifier 
• Example:
Let the u.d. of x be parking spaces at BU.
Let P(x) be the predicate “x is full.”
• Then the universal quantification of P(x),
• x P(x), is the proposition:
• “All parking spaces at BU are full.”
• “Every parking space at BU is full.”
• “For each parking space at BU, that space is full.”

8. The Existential Quantifier 
• Example:
Let the u.d. of x be parking spaces at BU.
Let P(x) be the predicate “x is full.”
• Then the existential quantification of P(x),
x P(x), is the proposition:
• “There is a parking space at BU that is full.”
• “At least one parking space at BU is full.”
• “Some parking spaces at BU is full.”

9. Quantifier Equivalence Laws
• Definitions of quantifiers: If u.d.=a,b,c,…
x P(x) ≡ P(a)  P(b)  P(c)  …
x P(x) ≡ P(a)  P(b)  P(c)  …
• x y P(x,y) ≡ y x P(x,y)
x y P(x,y) ≡ y x P(x,y)
• x (P(x)  Q(x)) ≡ (x P(x))  (x Q(x))
x (P(x)  Q(x)) ≡ (x P(x))  (x Q(x))

10. ThankYou