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ICS 6D Logic Overview Instructor: Sandy Irani
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What is logic? Branch of mathematics in which variables and expressions have value true (T) or false (F) Particularly relevant to CS: – AI: automated reasoning – Digital logic design Useful in any area in which it is important to make precise statements and reason in a systematic way.
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Proposition A proposition is a statement that is either true or false: – 2 + 3 = 5 – 3 + 4 = 6 – 4 is a prime number – It is raining today – It will rain tomorrow (unknown truth value) – Chocolate is the best flavor of ice cream (matter of opinion)
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Propositions Statements that are not propositios: – How are you doing? (Question) – Have a nice day! (Command) – Bummer! (exclamation) Propositional variables – Typically: p, q, r, s – Have value T or F.
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Logical Operations: Conjunction Conjunction (AND) – p ˄ q is true when p is true and q is true, false otherwise Truth table shows the truth value of an expression for every possible combination of truth values for the individual propositions In the expression. pq p˄qp˄q
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Logical Operations: Disjunction Disjunction (OR) – p ˅ q is true when p is true or q is true or both are true. p ˅ q only when p and q are both false. pq p ˅ q
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Inclusive vs. Exclusive OR Inclusive OR (True if either or both propositions are true) – The patient should not take the medication if she has a history of migraines or she has diabetes. Exclusive OR (True if exactly one of the propositions is true. False if both propositions are true) – I will drive or ride my bike to work today. In logic: “OR” is always the inclusive OR unless explicitly stated otherwise.
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Logical Operations: Complement Complement (NOT) p p pp
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Compound Propositions p q r Order of precedence: – Complement – Conjunction – Disjunction Good to use parens: – Except for complement, multiple , multiple
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Compound Propositions p: true q: false r: true p (q r) (p q r)
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Truth table for: p (q r) p (q r) pqr pp rrq r
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Logical Equivalence Two compound propositions are logically equivalent if they have the same truth value for every combination of truth values for their individual propositions. (p q) p q pq (p q) p q
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De Morgan’s Law: (p q) p q p: The applicant is over 18 years old q: the applicant has a valid driver’s license (p q): It is not true that the applicant is over 18 years old and has a valid driver’s license. p q: The applicant is not over 18 years old or does not have a valid driver’s license.
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De Morgan’s Law: (p q) p q p: the patient has a history of migraines q: the patient has diabetes (p q): It is not true that the patient has a history of migrains or has diabetes. p q: The does not have a history of migraines and does not have diabetes.
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The conditional operation “Implies” p → q – p is the “hypothesis” and q is the “conclusion” pqp → q “If p, then q.” “q, if p.” “p implies q”
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The conditional operation h: you mow my lawn c: I pay you $20 h → c: If you mow my lawn, I will pay you $20. You mow my lawn You don’t mow my lawn I pay you $20 I don’t pay you $20
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The conditional operation s: Suzy studied for her test p: Suzy passed her test p → q p q: – If Suzy studied for her test then she passed her test. – Either Suzy did not study for her test or she passed her test.
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Contrapositive s: Suzy studied for her test p: Suzy passed her test The contrapositive of p → q is q → p. p → q q → p – If Suzy studied for her test then she passed her test. – If Suzy did not pass her test, then she did not study for it.
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The bi-conditional operation “If and only if” p ↔ q – p ↔ q is true whenever p and q have the same truth value. pqp ↔ q
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Convenient logical equivalences p → q p q De Morgans Laws: (p q) p q (p q) p q Associative Laws: (p q) r p (q r ) (p q) r p (q r )
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Disjunction/Conjunction p 1 p 2 p 3 p 4 …. p n p 1 p 2 p 3 p 4 …. p n
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Predicate P(x): “x is prime” is a predicate (not a proposition) – Truth value depends on the value of x. – P(5): “5 is prime.” P(5) is a proposition. – P(6): “6 is prime.” P(6) is a proposition. Variable x in a predicate has a domain – Set of possible values for x. – For “x is prime” a natural domain is the set of positive integers.
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Predicate Domain: set of students in a class: – {Abigail, Bernice, Charlie, …, Zachary} Predicate T(x): x got an A on the test. – T(Bernice): “Bernice got an A on the test.” – T(Charlie) T(Zachary): “Charlie and Zachary both got an A on the test.” – T(Zachary) → T(Bernice): “If Zachary got an A on the test then Bernice got an A on the test.” (All propositions.)
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Existential Quantifier Domain: set of students in a class: – {Abigail, Bernice, Charlie, …, Zachary} Predicate T(x): x got an A on the test. – ∃ x T(x) “There is a student who got an A on the test.” “Some student got and A on the test.” “At least one student got an A on the test.” For finite domains: – ∃ x T(x) ( T(Abigail) T(Bernice) T(Charlie) … T(Zachary))
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Existential Quantifier ∃ x T(x) True if T(n) is true for at least one value for x in the domain. False if T(n) is false for every value of x in the domain. Example: domain is the set of all positive integers. ∃ x (x 2 = x) ∃ x (x + x = x)
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Universal Quantifier Domain: set of students in a class: – {Abigail, Bernice, Charlie, …, Zachary} Predicate P(x): x passed the test. – ∀ x P(x) “Every student passed the test.” “All students passed the test.” For finite domains: – ∀ x P(x) ( P(Abigail) P(Bernice) P(Charlie) … P(Zachary))
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Universal Quantifier ∀ x T(x) True if T(n) is true for every value for x in the domain. False if T(n) is false for at least one value for x in the domain. (counterexample) Example: domain is the set of all positive integers. ∀ x (x 2 = x) ∀ x (x 2 ≥ x) ∀ x (x 2 > x)
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Universal Quantifier Examples P(x): x is prime. D(x): x is odd The domain for x is the set of all positive integers ∀ x (P(x) → D(x)) ∀ x (P(x) → (D(x) (x = 2)) ) ∀ x ((x < 0) → P(x))
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