Midterm• Monday, October 5 in class (1:40pm-3:00pm)

• Here in Lucy Stone Hall Auditorium

• Will announce additional office hours

Midterm: Rules

• Bring valid Photo ID, pens.

• Closed book

• One sheet (front and back) of handwritten or typed notes (no photocopies) which instructor and TAs need to be able to read without any aids.

• No electronics (includes watches, calculators, cell phones, tablets, computers)

Midterm: Rules

• Switch off your cell phone!

• Leave all backpacks (with all electronics turned off), jackets and other bulky items in front of room

• Take only pen, pencil, eraser, Photo ID, and something to drink to your seat

• If you finish your exam within 60 minutes, you may hand it in and leave. Otherwise wait until the end.

Midterm: Scope

• Book: Chap 1.1-1.5 and Chap 2

• HW 1-3

• Clicker problems up to and including 9/23

Tentative Grading

• Weekly assignments: 20%

• Two midterms: 30%

• Comprehensive Final: 40%

• iClicker Participation: 10% Note: iClicker grade will be based on being significantly better than random guessing

• Active participation in recitations is used to bump up grade or make up for other issues

Exam grading

All Data from CS206 in Fall 2014

Exam grading on a curve

Midterm: Strategy

• Don’t Panic!

• Don’t expect to solve all problems (even if 4.0 student).

• Read through complete exam and triage problems

• Points roughly correspond to difficulty

• No trick questions: ask if anything is unclear

QuantifiersName Notation When true? When false

Universal Quantification

∀x P(x)“for all”

P(x) is true for all x

There is an x for which P(x) is false (counterexample)

ExistentialQuantification

∃x P(x)“exists”

There is an x for which P(x) is

true

P(x) is false for all x

De Morgan’s for Quantifier

Name Notation When is negation true?

When is negation false?

¬∃xP(x) ∀x ¬ P(x) P(x) is false for every x

There is an x for which P(x) is true

¬∀x P(x)∃x ¬P(x) There is an x for

which P(x) is false

P(x) is true for all x

Consider the domain R+. Which of the propositions is the negation of ∀x∃y(x+y=1)

Negating Nested Quantifiers

A ∀x∃y(x+y ≠ 1)

B ∃x∃y(x+y ≠ 1)

C ∀x∀y(x+y ≠ 1)

D ∃x∀y(x+y ≠ 1)

E none of the above

D)

Example of an Argument

• p “a given number is divisible by 4”

• q “a given number is divisible by 2”

• Premises:

• p

• p → q

• Conclusion q is true when all premises true. The argument is valid.

Example of an Argument Form

• p “a given number is divisible by 4”

• q “a given number is divisible by 2”

• Premises:

• p

• p → q

• Conclusion q is true when all premises true. The argument form is valid for all p, q.

Example of an Argument Form

• Premises:

• p

• p → q

• Conclusion q is true when all premises true. The argument form is valid for all p, q.

p→ qp

∴q

Modus ponens

General Argument Form

• Premises p1, p2, …, pn

• The argument form is valid for all premises p1, p2, …, pn if conclusion q is true when all premises p1, p2, …, pn true; i.e.,if (p1 ∧ p2 ∧…∧ pn)→q is a tautology.

Note

• (p1 ∧ p2 ∧…∧ pn)→q is a tautology does not mean q is always true

• q “a number is divisible by 4”

• p “a number is divisible by 2”

• Conclusion q is not true because p→q is not true. The argument remains valid.

p→ qp

∴q

(p∧(p→q))→q Modus ponens

(¬q∧(p→q))→¬p Modus tollens

((p→q)∧(q→r))→(p→r)Hypothetical

syllogism

((p∨q)∧¬p)→qDisjunctive syllogism

p→(p∨q) Addition

(p∧q)→ p Simplification

((p)∧(q))→ (p∧q) Conjunction

((p∨q)∧(¬p∨r))→(q∧r) Resolution

p→ qp

∴q¬qp→ q

∴¬pp→ qq→ r

∴ p→ qp∨ q¬p

∴qp∴ p∨ qp∧ q∴ ppq

∴ p∧ qp∨ q¬p∨ r

∴q∧ r

• p “a given number is divisible by 4”

• q “a given number is divisible by 2”

• Premises:

• ¬q

• p → q

• Conclusion ¬p is true by ?

A B C D E

Modusponens

Hypothetical syllogism

Disjunctive syllogism

Modustollens

B)

• p “a given number is divisible by 4”

• q “a given number is divisible by 2”

• r “a given number is divisible by 8”

• Premises:

• p → q, r →p

• Conclusion r→q is true by ?

A B C D E

Modusponens

Hypothetical syllogism

Disjunctive syllogism

Modustollens

C)

Universal Instantiation

Universal Generalization

Existential Instantiation

Existential Generalization

∀xP(x)∴P(c)P(c) for an arbitrary c∴∀xP(x)∃xP(x)∴P(c) for some element c

P(c) for some element c∴∃xP(x)

Universal Modus Ponens

Universal Modus Tollens

∀x(P(x)→Q(x))P(c) where c is a particular element in domain

∴Q(c)

∀x(P(x)→Q(x))¬Q(c) where c is a particular element in domain

∴¬P(c)

Argument w/ Propositional Functions

• P(x) “the number x is divisible by 4”

• Q “the number x is divisible by 2”

• Premises:

• P(c)

• ∀x(P(x) → Q(x))

• Conclusion Q(c) is true when all premises true. The argument is valid.

Example

• “If I take the day off, it either rains or snows”

• “I took Tuesday off or I took Thursday off”

• “It was sunny on Tuesday”

• “It did not snow on Thursday”

• “If I take the day off, it either rains or snows”

• “I took Tuesday off or I took Thursday off”

• “It was sunny on Tuesday”

• “It did not snow on Thursday”

• P(x) took day x off

• Q(x) day x sunny

• R(x) day x rainy

• S(x) snow on day x

1. ∀x(P(x) → (R(x) ∨ S(x)))

2. P(Tu) ∨ P(Th)

3. Q(Tu)

4. ¬S(Th)

Premises