This document introduces predicates and quantifiers in predicate logic. It defines predicates as functions that take objects and return propositions. Predicates allow reasoning about whole classes of entities. Quantifiers like "for all" (universal quantifier ∀) and "there exists" (existential quantifier ∃) are used to make general statements about predicates over a universe of discourse. Examples demonstrate how predicates and quantifiers can represent concepts like "all parking spaces are full" or "some parking space is full." Laws of quantifier equivalence and negation rules with quantifiers are also presented.

1. Predicates and Quantifiers
Discrete Mathematics
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2. Slide Owner
• Istiak Ahmed
• Email- istiakahmed271@gmail.com
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3. 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).
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4. 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.
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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.
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6. Subjects and Predicates
• Example “The dog is sleeping”:
• In predicate logic, a predicate is modeled as a function P(
) from objects to propositions.
• P(x) = “x is sleeping” (where x is any object).
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7. More About Predicates
• Convention: Lowercase variables x, y, z... denote
objects/entities; uppercase variables P, Q, R… denote
propositional functions (predicates).
• The result of applying a predicate P to an object x is the
proposition P(x).
• The predicate P itself (e.g. P =“is sleeping”) is not a
proposition (not a complete sentence).
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8. Propositional Functions
• Predicate logic can also involve statements with more
than one variable or argument.
• E.g. let P(x,y,z) = “x gave y the grade z”, then if
x=“Mike”, y=“Mary”, z=“A”, then P(x,y,z) = “Mike gave
Mary the grade A.”
• E.g. let Q(x,y,z) = “x + y > z”, then what is the truth value of
Q(1,2,3)? – False
• Q(3,2,1)? -True
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9. Universes of Discourse
• A mathematical function may be valid for all values of
a variable in a particular domain, called the universe
of discourse.
• E.g., let P(x)=“x+1>x”. We can then say,
“For any number x, P(x) is true” instead of
(0+1>0)  (1+1>1)  (2+1>2)  ...
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10. Quantifier Expressions
• “” is the FOR LL or universal quantifier.
x P(x) means “P(x) is true for all values of x in the universe
of discourse”.
• “” is the XISTS or existential quantifier.
x P(x) means “there exists an x in the u.d. (that is, 1 or
more) such that P(x) is true”.
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11. 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.”
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12. 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.”
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13. 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))
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14. Negation Example
• x P(x) ≡ x P(x)
• P(x)=“x is a student of cse” where u.d. is the students of this
class
• So, x P(x) = “All students in this class is a student of cse”
• The negation of this x P(x) = “Not every student in this
class is a student of cse”
• This is the same as x P(x) = “There is a student x who is
not a student of cse”
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15. Thanks To All
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