Lecture 1.3: Predicate Logic

1. Lecture 1.3: Predicate Logic CS 250, Discrete Structures, Fall 2015 Nitesh Saxena Adopted from previous lectures by Cinda Heeren

2. Course Admin • Slides from last lectures all online • Both ppt and pdf • Any questions? Lecture 1.3 - Predicate Logic

3. Outline • Predicate Logic (contd.) Lecture 1.3 - Predicate Logic

4. Quantifiers – another way to look at them • To simplify, let us say that the universe of discourse is {x1, x2 } • x P(x)  P(x1)  P(x2) • x P(x)  P(x1)  P(x2) • This is very useful in proving equivalences involving propositions that use quantifiers • Let us see some examples Lecture 1.3 - Predicate Logic

5. Laws and Quantifiers • Negation or De Morgan’s Law (we saw this last time): • x P(x)  x P(x) • x P(x)  x P(x) • Distributivity: • x (P(x)  Q(x))  x P(x)  x Q(x) • x (P(x)  Q(x))  x P(x)  x Q(x) • Can’t distribute universal quantifier over disjunciton or existential quantifier over conjunction Lecture 1.3 - Predicate Logic

6. Reminder: in a proposition, all variables must be bound. Predicates – Free and Bound Variables A variable is bound if it is known or quantified. Otherwise, it is free. Examples: P(x) x is free P(5) x is bound to 5 x P(x) x is bound by quantifier Lecture 1.3 - Predicate Logic

7. True proposition • False proposition • Not a proposition • No clue c) b) b) b) Predicates – Nested Quantifiers To bind many variables, use many quantifiers! Example: P(x,y) = “x > y”; universe of discourse is natural numbers • x P(x,y) • xy P(x,y) • xy P(x,y) • x P(x,3) Lecture 1.3 - Predicate Logic

8. P(x,y) true for all x, y pairs. For every value of x we can find a y so that P(x,y) is true. P(x,y) true for at least one x, y pair. There is at least one x for which P(x,y) is always true. 1 and 2 are commutative 3 and 4 are not commutative Suppose P(x,y) = “x’s favorite class is y.” Predicates – Meaning of Nested Quantifiers • xy P(x,y) • xy P(x,y) • xy P(x,y) • xy P(x,y) Lecture 1.3 - Predicate Logic

9. False True True False Nested Quantifiers – example N(x,y) = “x is sitting by y” • xy N(x,y) • xy N(x,y) • xy N(x,y) • xy N(x,y) Lecture 1.3 - Predicate Logic

10. Today’s Reading and Next Lecture • Rosen 1.5 • Again, please start solving the exercises at the end of each chapter section! • Please read 1.6 and 1.7 in preparation for the next lecture Lecture 1.3 - Predicate Logic