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Logic Proofs

Monday 9/29/08

- Take out HW Drawing Conclusions Pick up

handouts! - Quiz handed back Wednesday
- Logic Proofs
- - Law of Detachment
- HW 1. Essay Due Thursday!
- 2. Law of Detachment

Definitions

- Logically equivalent when two statements always

have the same truth value. - Premise the statement that is given and

excepted to be true. - Conclusion the statement that has come from the

premises.

Law of Detachment(a law of inference)

- The Law of Detachment states that when two given

premises are true, one a conditional and the

other the hypothesis of that conditional it then

follows that the conclusion of the conditional is

true. - If you are given a conditional and the hypothesis

the conclusion is true. - p? q and p therefore q

End for today

Wednesday 10/1/08

- Take out HW Law of Detachment Pick up

handout! - Place 1 4 answers (truth tables) on board!

HW pass - 2. Discuss Quiz
- 3. Logic Proofs
- - The Law of Contrapositive
- - Formal Proof
- - Modus Tollens
- HW 1. Essay Due Thursday!
- 2. Law of Contrapositive / Modus Tollens

Law of Contrapositive

- The Law of Contrapositive states that when a

conditional premise is true, it follows that the

contrapositive of the premise is also true. - If the conditional is true the contrapositive is

also true. - p ? q then q ? p

- Formal Proof given premises that are true use

laws of reasoning to reach a conclusion. - Two Column Proof
- Column 1 Statements
- Column 2 Reasons
- Each statement must have a reason and they are

number in sequence.

Example

- Given If Joanna saves enough money, then she

can buy a bike. Joanna can not buy a bike. - Prove Joanna did not have enough money.
- Let m Joanna saves enough money
- Let b Joanna can buy a bike
- Given m ? b, b
- Prove m

- Reason
- Given
- Given

- Statement
- m ? b
- b

3. Law of Contrapositive (step 1)

3. b ? m

4. m

4. Law of Detachment (step 2 3)

End for today

Quiz Review

1. Rewrite the following as an equivalent

DISJUNCTION!

a. 2x ? 4

a. (2x lt 4) ? (2x gt 4)

b. (b - 2 lt 2) ? (b - 2 gt 2)

b. b - 2 ? 2

- Write the negation of each in simplest terms!
- A. 10 x lt 5
- B. I never fail quizzes.
- C. Math is logical.
- D. It is not true that I am a good baseball

player.

A. 10 x 5

B. I sometimes fail quizzes

C. Math is not Logical.

D. I am a good baseball player.

Quiz Review

- Write the following in symbolic form using the

letters given. In each case the letter is the

positive value - of the statement!

a. The absolute value of x is equal to

2 if and only if x 2 or x -2. (p

absolute value of x is 2 q x 2 r x -2)

b. If Im late, then Ill get into trouble, and

if Im not late, then I wont get into

trouble.. (l,t)

Answers placed on board!

Thursday 10/2/08

- Take out HW Law of Contrapositive Place

essay in box! - Complete Pre-game Warm-up! See Quiz Review
- 3. Logic Proofs
- - Modus Tollens
- - Chain Rule (Law of Syllogism)
- HW 1. Law of Contrapositive / Modus Tollens

21-30 - 2. Complete Chain Rule Worksheet

Law of Modus Tollens

- Given If the tickets are sold out (t),

then well wait for the next show. (w) - We do not wait for the next show.
- Prove The tickets were not sold out.

Write the problem in symbolic notation!

Law of Modus Tollens

- Given t ? w t ? w
- w w
- Prove t ? t

or (t ? w) ? w ? t

Set up a truth table to prove!

Prove (t ? w) ? w ? t

Prove (t ? w) ? w ? t

(t ? w) ? w ? t is a Tautology therefore a

valid argument!

Law of Modus Tollens

- The law of Modus Tollens states when 2 given

premises are true, one a conditional and the

other the negation of the conclusion of that

conditional, it then follows that the negation of

the hypothesis of the conditional is true. - Given a conditional and the negation of the

conclusion then the negation of the hypothesis is

true. (combining Law of Contrapositive and Law

of Detachment) - p ? q, q therefore p

Invalid Arguments

- All premises are true but they do not all lead to

a valid argument. - Example p?q
- q
- no conclusion
- p?q
- p
- no conclusion

End for Today

Friday 10/3/08

- Take out HW Law of Contrapositive, Chain Rule

worksheet - Complete Quiz Retake! See Quiz Review handout
- 3. Logic Proofs
- - Discuss Valid arguments
- - Chain Rule (Law of Syllogism) Applications

of chain rule - - Law of Disjunctive Inference
- HW 1. Law of Disjunctive Inference worksheet
- 2. Application of Chain Rule 2-40 (evens)

Quiz Review

1. Rewrite the following as an equivalent

DISJUNCTION!

-x 5 ? -3

- Write the negation of each in simplest terms!
- A. x -5
- B. Monkeys do not live in trees.
- C. Math is never logical.

Quiz Review

- Write the following in symbolic form using the

letters given. In each case the letter is the

positive value - of the statement!

If I dont go to practice and train hard, then I

will not be prepared for the game next week and

our team will not win (p,h,g,w)

Chain Rule (Law of Syllogism)

(p ? q) ? (q ? r) ? (p ? r)

Chain Rule (Law of Syllogism)

(p ? q) ? (q ? r) ? (p ? r)

Chain Rule (Law of Syllogism)

(p ? q) ? (q ? r) ? (p ? r)

Valid Arguments

- An argument is valid if the implication (P1 P2

P3 P4 . Pn) C.

?

?

?

?

?

?

Premises

Premises

Is a valid

argument?

(p ? q) ? q ? p

Law of Modus Tollens

Chain RuleLaw of Syllogism

- Chain rule states that when 2 given premises are

true conditional such that the consequent

(conclusion) of the 1st is the antecedent

(hypothesis) of the 2nd it follows that a

conditional formed using the antecedent of the

1st and the consequent of the 2nd is true. - You can combine conditionals if the conclusion of

one is the hypothesis of another. - p ? q and q? r then p ? r

Chain RuleLaw of Syllogism

p ? q

q ? r

? p ? r

(p ? q) ? (q ? r) ? (p ? r)

Chain RuleExample

p You study

q You pass

r You get a surprise

P1

p ? q

If you study, then you will pass.

P2

q ? r

If you pass, then you will get a surprise.

Chain RuleExample

p You study

q You pass

r You get a surprise

P1

p ? q

If you study, then you will pass.

P2

q ? r

If you pass, then you will get a surprise.

? p ? r

C

If you study, then you will get a surprise.

Law of Disjunction

- The previous laws involved conditionals this one

does not. - Law of Disjunctive Inference states when 2 given

premises are true, on a disjunction and the other

the negation of one of the disjuncts it then

follows the other disjunct is true. - p ? q or p ? q
- q p

? p

? q

End for Today

Monday 10/6/08

- Take out HW Law of Disjuctive Inference, Chain

Rule worksheet Chain Rule Applications Place

proofs on the white board! - Another Proof - 10 Disjunctive Inference
- 3. Logic Proofs
- - Double Negation
- - DeMorgans Law
- 4. Begin HW Pass back Quizzes
- HW 1. pg 86-87 (evens)
- 2. Parent Signatures (3/2- by tomorrow!)

10 Hw Disjunctive Inference

Givens

Don is first or Nancy is Second. If Nancy is

second, then Chris is third. If Chris is third,

then Pattie is fourth. Pattie is not fouth.

Prove Pattie is not fourth.

Negations

- Contrapositive,
- p ?q ? q ?(p)
- p ?q ? q ? p
- Modus Tollens
- (p ?q) ? q ? (p)
- (p ?q) ? q ? p

Law of Double Negation

- Law of double negation states (p) and p are

logically equivalent. - (you do not need to apply this law, continue as

we have been)

DeMorgans Laws

- -Discovered by English mathematician
- -Tells us how to negate a conjunction and

disjunction - Complete the following truth table

Since (ms) and mVs are logically equivalent

for all cases, the negation of a conjunction is

the disjunction.

DeMorgans Law states

- The negation of a conjunction of 2 statements is

logically equivalent to the disjunction of the

negation of each of the two statements. - The negation of a disjunction of 2 statements is

logically equivalent to the conjunction of the

negation of each of the 2 statements. - (p q) ?? (p V q)
- (pV q ) ? ? (p q)

End for today

Laws of Simplification

Mike likes to read and play basketball. We can

conclude-

Mike likes to read. Mike likes to play

basketball.

Laws of Simplification

- The law of simplification states that when a

single conjunctive premise is true, it follows

that each of the individual conjucts must be

true. - p q therefore p is true
- p q therefore q is true

Law of Conjunction

- The law of conjunction states that when 2 given

premises are true, it follows that the

conjunction of these is true. - p
- q
- p q

Law of Conjunction

AB is perpendicular to CD

AB is the bisector of CD

AB is the perpendicular bisector of CD.

Law of Disjunctive Addition

- Law of disjunctive addition states that when a

single premise is true, it follows that any

disjunction that has this premise as a disjunct

is also true. - p
- p V q

Law of Disjunctive Addition

We solve an equation and see that x 5.

We may also include that the statement x

5 or x gt 5 is

True.

We may also include that the statement x

5 or Ian has 3 eyes is

True.

Other Problems

Given the statement r, which is a valid

conclusion

?

1. r k 2. r k 3. r k

4. r k

V

Wednesday 10/8/08

- Take out HW Law of Simplification, Place

proofs on the white board! - 2. Logic Proofs Practice!!!
- HW Logic Proof Quiz Friday!

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