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Negating Nested Quantifiers (xy xy 1) More examples: , =``student is enrolled in class ” 1. ¬(∃ ∀ , ) 2. ∃ ∃ ∀ ( ≠ ∧ , → , ) I(x)=“x has an internet connection” internet” Domain is students in your class. C(x,y)=“x and y have chatted over the 1. Someone in your class has an Internet connection but has not chatted with anyone else in the class. 2. There are two students in the class who between them have chatted with everyone else in the class. Section 1.5 – Rules of Inference • Terms: – Argument – Premises – Conclusion – Valid Standard Rules of Inference (Each is based on a tautology) Modus Ponens p pq q Modus Tollens q pq p Hypothetical Syllogism pq qr pr Standard Rules of Inference (Continued) Addition Simplification Conjunction p pq pq p p q pq Standard Rules of Inference (Continued) Disjunctive Syllogism pq p q pq Resolution p r q r Examples Alice is a mathematics major. Therefore, Alice is either a mathematics major or a computer science major. If it snows today, the university will close. The university is not closed today. Therefore, it did not snow today. If I go swimming, then I will stay in the sun too long. If I stay in the sun too long, then I will sunburn. Therefore, if I go swimming, then I will sunburn. Example Use rules of inference to show that the hypotheses “Randy works hard,” “If Randy works hard, then he is a dull boy” and “If Randy is a dull boy, then he will not get the job” imply the conclusion “Randy will not get the job.” Rules of Inference for Quantified Statements Universal Instantiation Universal Generalization x P( x) P(c) for any fixed value c P(c) for an arbitrarily chosen value c x P( x) Rules of Inference for Quantified Statements (Continued) Existential Instantiation Existential Generalization x P( x) P(c) for some valuec P(c) for some particularvalue c x P( x) Combining Rules of Inference for Quantified Statements Universal Modus Ponens Universal Modus Tollens x ( P( x) Q( x)) P (c ) Q (c ) x ( P( x) Q( x)) Q(c) P(c) Examples: Drawing Conclusions “Every computer science major has a personal computer.” “Ralph does not have a personal computer.” “Ann has a personal computer.” “Joe is a computer science major.” Valid Arguments vs Fallacies • Valid arguments are constructed using… • A fallacy is a (so-called) argument which is not so constructed. – Affirming the conclusion – Denying the hypothesis – Begging the question (( p q) q) p (( p q) p) q p Examples: Valid Argument or Fallacy? 1. All students in this class understand logic. Xavier is a student in this class. Therefore, Xavier understands logic. 2. Every computer science major takes discrete mathematics. Natasha is taking discrete mathematics. Therefore, Natasha is a computer science major. 3. All parrots like fruit. My pet bird is not a parrot. Therefore, my pet bird does not like fruit. 4. Everyone who eats granola every day is healthy. Linda is not healthy. Therefore, Linda does not eat granola every day. Section 1.6 – Introduction to Proofs Formal Proofs Definitions: Proof- Theorem- PropositionAxiom or postulate- Definitions Continued: Lemma- Corollary- Conjecture- Quantifiers • Remember that when no quantifier is given, a universal quantification is assumed. If xy > 0, then either x and y are both positive or x and y are both negative Some basic facts/definitions we’ll need: • An integer is even if there exists an integer such that = 2. • An integer is odd if there exists an integer such that = 2 + 1. • An integer is a perfect square if there is an integer such that = 2 . • If a and b are integers with ≠ 0, we say that divides if there is an integer such that = . • The real number is rational if there exist integers and with ≠ 0 such that = /. A real number that is not rational is called irrational. Methods of Proving → (Given arbitrarily complicated compound propositions p and q) Direct proof: Assume p is true. Show by a direct argument that q is true. Task: Prove the statement: “If a person likes math then he/she is cool.” Proof: Example: Prove by a direct argument that if is a perfect square then is either odd or divisible by 4. Methods of Proving → (Given arbitrily complicated compound propositions p and q) Indirect proof: Assume q is false. Show by a direct argument that p is false. Task: Prove the statement: “If a person likes math then he/she is cool.” Proof: Example: Prove by an indirect argument that if and are integers and is even, then either or must be even. Proving ↔ 1. Show that p→q 2. Show that q→p Task: Prove the statement: “A person likes math if and only if he/she is cool.” Proof: Proving Multiple Statements Equivalent Prove these statements are equivalent, where a and b are real numbers: (i) a is less than b, (ii) the average of a and b is greater than a, and (iii) the average of a and b is less than b. Other Types of Proof • Vacuous proof • Trivial proof • Proof by contradiction Prove that the product of a non-zero rational numbers and an irrational number is irrational using proof by contradiction. Mistakes in Proofs 1. 2. 3. 4. 5. 6. 7. = 2 = 2 − 2 = − 2 − + = − + = 2 = 2=1 Given Multiply both sides by a Subtract 2 from both sides Factor Divide by − Substitute for since = Divide both sides by b Section 1.7 – Proof Methods and Strategy • Proof by cases • Exhaustive Proof Prove that for any two real numbers and , + ≤ + . Theorems and Quantifiers • Existence proofs (constructive vs. nonconstructive) Constructive: Show that there is a positive integer that can be written as the sum of cubes of positive integers in two different ways. Nonconstructive: Prove that there exists two irrational numbers and for which is rational. Uniqueness quantifier and uniqueness proofs ∃! means Example: ∀, ≠ 0 → ∃! ( = 1). Counter-Examples • To show it is false that ∀ simply exhibit one value of for which () is false. • Example: Conjecture- Every positive integer is the sum of three squares. Open Problems The 3 + 1 conjecture: Starting with any positive integer and repeatedly applying the transformation whereby an even integer gets divided by 2 and an odd integer gets multiplied by 3 and incremented by 1, we will ultimately generate the integer 1. Goldbach’s conjecture: Every positive even integer n 4 can be expressed as the sum of two prime numbers.