Negating Nested Quantifiers

)1( xyyx

More examples: ``student is enrolled in class ”

1.

2.

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.

I(x)=“x has an internet connection” C(x,y)=“x and y have chatted over the internet”Domain is students in your class.

Section 1.5 – Rules of Inference

• Terms:– Argument

– Premises

– Conclusion

– Valid

Standard Rules of Inference (Each is based on a tautology)

Modus Ponens

Modus Tollens

Hypothetical Syllogismrp

rqqp

pqp

q

qqp

p

Standard Rules of Inference(Continued)

Addition

Simplification

Conjunction

qpqp

pqp

qpp

Standard Rules of Inference (Continued)

Disjunctive Syllogism

Resolution

rqrp

qp

qpqp

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.

ExampleUse 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

ccPxPx

valuefixedany for )()(

Universal

Instantiation

Universal Generalization )(

uechosen valy arbitrarilan for )(xPx

ccP

Rules of Inference for Quantified Statements (Continued)

ccPxPx

valuesomefor )()(

Existential

Instantiation

Existential Generalization )(

valueparticular somefor )(xPx

ccP

Combining Rules of Inference for Quantified Statements

)( )())()((

cQcP

xQxPx

Universal Modus Ponens

Universal Modus Tollens

)( )(

))()((

cPcQ

xQxPx

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

pqqp ))((

qpqp ))((

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 ProofsDefinitions:

Proof-

Theorem-

Proposition-

Axiom 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 .

• An integer is odd if there exists an integer such that .

• An integer is a perfect square if there is an integer such that .

• If a and b are integers with , we say that divides if there is an integer such that .

• The real number is rational if there exist integers and with 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 ifis 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

GivenMultiply both sides by aSubtract from both sidesFactorDivide 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 numbersand , .

Theorems and Quantifiers• Existence proofs (constructive vs. non-

constructive)

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:

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 ProblemsThe 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.