Discrete Mathematics

Show that [(p  q)  (q  r)] (p  r) is a tautology

b) Not everyone is perfect. c) All your friends are perfect. Let P (x) be "x is perfect"; let F(x) be "x is your friend"; Translate each of these statements into logical expressions using predicates, quantifiers, and logical connectives. a) No one is perfect. b) Not everyone is perfect. c) All your friends are perfect. d) At least one of your friends is perfect. e) Not everybody is your friend or someone is not perfect. x P(x) x P (x) x(F(x)  P (x)) x(F(x)  P (x)) (x F(x)  (xP(x))

Predicates - quantifier negation x (L(x)  H(x)) Not all large birds live on honey. x P(x) means “P(x) is true for every x.” What about x P(x) ? Not [“P(x) is true for every x.”] “There is an x for which P(x) is not true.” x P(x) So, x P(x) is the same as x P(x). x (L(x)  H(x)) 6/12/2018

Predicates - quantifier negation x (L(x)  H(x)) No large birds live on honey. x P(x) means “P(x) is true for some x.” What about x P(x) ? Not [“P(x) is true for some x.”] “P(x) is not true for all x.” x P(x) So, x P(x) is the same as x P(x). x (L(x)  H(x)) 6/12/2018

Predicates - quantifier negation So, x P(x) is the same as x P(x). So, x P(x) is the same as x P(x). General rule: to negate a quantifier, move negation to the right, changing quantifiers as you go. 6/12/2018

Predicates - quantifier negation Negate each of the following statements: (a) ∃x ∀y P(x, y) (b) ∃x ∀y P(x, y) (c) ∃y ∃x ∀z P(x, y, z) Use ￢∀x p(x) ≡ ∃x￢p(x) and ￢∃x p(x) ≡ ∀x￢p(x): (a) ￢(∃x∀y, p(x, y)) ≡ ∀x∃y￢p(x, y) (b) ￢(∀x∀y, p(x, y)) ≡ ∃x∃y￢p(x, y) (c) ￢(∃y ∃x ∀z, p(x, y, z)) ≡ ∀y ∀x ∃z￢p(x, y, z) 6/12/2018

Predicates - free and bound variables A variable is bound if it is known or quantified. Otherwise, it is free. Examples: P(x) x is free P(5) x is bound to 5 x P(x) x is bound by quantifier Reminder: in a proposition, all variables must be bound. 6/12/2018

Predicates - multiple quantifiers To bind many variables, use many quantifiers! Example: P(x,y) = “x > y” x P(x,y) xy P(x,y) xy P(x,y) x P(x,3) c) b) True proposition False proposition Not a proposition No clue a) b) 6/12/2018

Predicates - the meaning of multiple quantifiers xy P(x,y) xy P(x,y) xy P(x,y) xy P(x,y) P(x,y) true for all x, y pairs. P(x,y) true for at least one x, y pair. For every value of x we can find a (possibly different) y so that P(x,y) is true. There is at least one x for which P(x,y) is always true. Suppose P(x,y) = “x’s favorite class is y.” quantification order is not commutative. 6/12/2018

Proofs - how do you know? The following statements are true: If I am Mila, then I am a great swimmer. I am Mila. What do we know to be true? I am a great swimmer! How do we know it? 6/12/2018

Axiom, postulates, hypotheses and previously proven theorems. Proofs - how do you know? A theorem is a statement that can be shown to be true. A proof is the means of doing so. Axiom, postulates, hypotheses and previously proven theorems. Rules of inference Proof 6/12/2018

What rule of inference can we use to justify it? Proofs - how do you know? The following statements are true: If I am Mila, then I am a great swimmer. I am Mila. What do we know to be true? I am a great swimmer! What rule of inference can we use to justify it? 6/12/2018

Proofs - Modus Ponens  q I am Mila. If I am Mila, then I am a great swimmer.  I am a great swimmer! p p  q  q Inference Rule: Modus Ponens Tautology: (p  (p  q))  q 6/12/2018

Proofs - Modus Tollens  p I am not a great skater. If I am Erik, then I am a great skater.  I am not Erik! q p  q  p Inference Rule: Modus Tollens Tautology: (q  (p  q))  p 6/12/2018

Proofs - Addition I am not a great skater.  I am not a great skater or I am tall. p  p  q Inference Rule: Addition Tautology: p  (p  q) 6/12/2018

Proofs - Simplification I am not a great skater and you are sleepy.  you are sleepy. p  q  p Inference Rule: Simplification Tautology: (p  q)  p 6/12/2018

Proofs - Disjunctive Syllogism I am a great eater or I am a great skater. I am not a great skater.  I am a great eater! p  q q  p Inference Rule: Disjunctive Syllogism Tautology: ((p  q)  q)  p 6/12/2018

Proofs - Hypothetical Syllogism If you are an athlete, you are always hungry. If you are always hungry, you have a snickers in your backpack.  If you are an athlete, you have a snickers in your backpack. Inference Rule: Hypothetical Syllogism Tautology: ((p  q)  (q  r))  (p  r) p  q q  r  p  r 6/12/2018

Proofs - A little quiz… Amy is a computer science major.  Amy is a math major or a computer science major. Addition If Ernie is a math major then Ernie is geeky. Ernie is not geeky!  Ernie is not a math major. Modus Tollens 6/12/2018

Proofs - A little proof… Here’s what you know: Ellen is a math major or a CS major. If Ellen does not like discrete math, she is not a CS major. If Ellen likes discrete math, she is smart. Ellen is not a math major. Can you conclude Ellen is smart? M  C D  C D  S M 6/12/2018

Proofs - A little proof… 1. M  C Given 2. D  C Given 3. D  S Given 4. M Given 5. C DS (1,4) 6. D MT (2,5) 7. S MP (3,6) Ellen is smart! 6/12/2018

Proofs - A little proof…(Another approach) 1. M  C Given 2. D  C Given 3. D  S Given 4. M Given 5. C Disjunctive Syllogism (1,4) 6. C  D Contrapositive of 2 7. C  S Hypothetical Syllogism (6,3) 8. S Modus Ponens (5,7) Ellen is smart! 6/12/2018

p be the proposition "It is sunny this afternoon," q the proposition "It is colder than yesterday," r the proposition "We will go swimming," s the proposition "We will take a canoe trip," and t the proposition "We will be home by sunset." Challenge.. Here’s what you know: "It is not sunny this afternoon and it is colder than yesterday" "We will go swimming only if it is sunny" "If we do not go swimming, then we will take a canoe trip" "If we take a canoe trip, then we will be home by sunset“ Can you conclude "We will be home by sunset." p  q r  p  r  s s  t t

p be the proposition "You send me an e-mail message," q the proposition "I will finish writing the program," r the proposition "I will go to sleep early," and s the proposition "I will wake up feeling refreshed." Challenge.. Here’s what you know: "If you send me an e-mail message, then I will finish writing the program," "If you do not send me an e-mail message, then I will go to sleep early," "If I go to sleep early, then I will wake up feeling refreshed" Can you conclude "If I do not finish writing the program, then I will wake up feeling refreshed." p  q  p  r r  s  q  s

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