Discrete Mathematics

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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