Logic Proofs

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Logic Proofs
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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

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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.

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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
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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

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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

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• 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.

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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

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• 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
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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.
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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!
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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

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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!
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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!
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Prove (t ? w) ? w ? t
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Prove (t ? w) ? w ? t
(t ? w) ? w ? t is a Tautology therefore a
valid argument!
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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

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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
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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)

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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.

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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)
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Chain Rule (Law of Syllogism)
(p ? q) ? (q ? r) ? (p ? r)
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Chain Rule (Law of Syllogism)
(p ? q) ? (q ? r) ? (p ? r)
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Chain Rule (Law of Syllogism)
(p ? q) ? (q ? r) ? (p ? r)
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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
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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

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Chain RuleLaw of Syllogism
p ? q
q ? r
? p ? r
(p ? q) ? (q ? r) ? (p ? r)
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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.
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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.
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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
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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!)

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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.
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Negations

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

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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)

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DeMorgans Laws

• -Discovered by English mathematician
• -Tells us how to negate a conjunction and
  disjunction
• Complete the following truth table

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Since (ms) and mVs are logically equivalent
for all cases, the negation of a conjunction is
the disjunction.
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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
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Laws of Simplification
Mike likes to read and play basketball. We can
conclude-
Mike likes to read. Mike likes to play
basketball.
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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

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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

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Law of Conjunction
AB is perpendicular to CD
AB is the bisector of CD
AB is the perpendicular bisector of CD.
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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

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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.
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Other Problems
Given the statement r, which is a valid
conclusion
?
1. r k 2. r k 3. r k
4. r k
V

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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!