Predicates and Quantifiers
- 1. Predicates and Quantifiers Discrete Mathematics Page 1 1
- 2. Slide Owner • Istiak Ahmed • Email- [email protected] Page 2 2
- 3. Propositional Logic • Atomic propositions: p, q, r, … • Boolean operators: • Compound propositions: (p q) r • Equivalences: pq ≡ (p q) • Proving equivalences using: • Truth tables. • Symbolic derivations (Laws). Page 3 3
- 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. Page 4 4
- 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. Page 5 5
- 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). Page 6 6
- 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). Page 7 7
- 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 Page 8 8
- 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) ... Page 9 9
- 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”. Page 10 10
- 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.” Page 11 11
- 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.” Page 12 12
- 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)) Page 13 13
- 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” Page 14 14
- 15. Thanks To All Page 15 15