Introduction to Logic

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1 Introduction to Logic

2 Why study Logic? Understand meaning of mathematical sentences. Develop the building blocks for mathematical reasoning. Write correct proofs for mathematical statements. Identify buggy proofs. Because it is fun!

3 Why study Logic? Understand meaning of mathematical sentences. Develop the building blocks for mathematical reasoning. Write correct proofs for mathematical statements. Identify buggy proofs. Because it is fun! Design algorithms / programs and prove that are correct and fast

4 What is a proof? Definition A mathematical proof of a proposition is a seq. of logical deductions leading to the proposition from a base set of premises/ axioms.

5 What is a proof? Definition A mathematical proof of a proposition is a seq. of logical deductions leading to the proposition from a base set of premises/ axioms. NOT a proof!

6 Propositions Definition A proposition is a declarative statement that is either true or false but not both.

7 Propositions Definition A proposition is a declarative statement that is either true or false but not both. Examples: 1. Today is a Thursday. 2. Chennai is the capital of India. 3. All students in CS1200 are from Tamil Nadu. 4. Do you like this course? 5. Bring me a glass of water. 6. x + 3 = =

8 Propositions Definition A proposition is a declarative statement that is either true or false but not both. Examples: 1. Today is a Thursday. 2. Chennai is the capital of India. 3. All students in CS1200 are from Tamil Nadu. 4. Do you like this course? not declarative 5. Bring me a glass of water. not declarative 6. x + 3 = 10. neither true nor false =

9 Are propositions alone sufficient? We encounter complicated statements like... Laziness is not good. It is hot yet pleasant today. I will have coffee or tea. If I read Rosen, I will score good marks. I will win the game only if I have practised earlier or my opponent is Stanley.

10 Compound propositions Propositional variables: p : It is hot today. q : It is pleasant today. r : I will have coffee. s : I will have tea.

11 Compound propositions Propositional variables: p : It is hot today. q : It is pleasant today. r : I will have coffee. s : I will have tea. Logical connectives: NOT : p (negation) p : It is not hot today.

12 Compound propositions Propositional variables: p : It is hot today. q : It is pleasant today. r : I will have coffee. s : I will have tea. Logical connectives: NOT : p (negation) p : It is not hot today. OR : p q (disjunction) p r : It is hot today or I will have coffee.

13 Compound propositions Propositional variables: p : It is hot today. q : It is pleasant today. r : I will have coffee. s : I will have tea. Logical connectives: NOT : p (negation) p : It is not hot today. OR : p q (disjunction) p r : It is hot today or I will have coffee. r s (are we willing to drink both tea and coffee?)

14 Compound propositions Propositional variables: p : It is hot today. q : It is pleasant today. r : I will have coffee. s : I will have tea. Logical connectives: NOT : p (negation) p : It is not hot today. OR : p q (disjunction) p r : It is hot today or I will have coffee. r s (are we willing to drink both tea and coffee?) many times in English we mean exclusive OR

15 Compound propositions Propositional variables: p : It is hot today. q : It is pleasant today. r : I will have coffee. s : I will have tea.

16 Compound propositions Propositional variables: p : It is hot today. q : It is pleasant today. r : I will have coffee. s : I will have tea. Logical connectives: XOR : p q p q : It is hot or pleasant today but not both. AND : p q (conjunction) p r : It is hot and pleasant today.

17 Compound propositions Propositional variables: p : It is hot today. q : It is pleasant today. r : I will have coffee. s : I will have tea. Logical connectives: XOR : p q p q : It is hot or pleasant today but not both. AND : p q (conjunction) p r : It is hot and pleasant today. It is hot yet pleasant today.

18 Compound propositions Propositional variables: p : It is hot today. q : It is pleasant today. r : I will have coffee. s : I will have tea. Logical connectives: XOR : p q p q : It is hot or pleasant today but not both. AND : p q (conjunction) p r : It is hot and pleasant today. It is hot yet pleasant today. many times in English we use but / yet instead of and

19 Conditional statement p q If the machine breaks down within a year, the company will replace it. p : Machine breaks down within a year. q : The company will replace it.

20 Conditional statement p q If the machine breaks down within a year, the company will replace it. p : Machine breaks down within a year. q : The company will replace it. Truth Table p q p q T T T T F F F T T F F T

21 Conditional statement p q If the machine breaks down within a year, the company will replace it. p : Machine breaks down within a year. q : The company will replace it. Truth Table p q p q T T T T F F F T T F F T How do you express p q in terms of logical connectives?

22 Conditional statement p q If the machine breaks down within a year, the company will replace it. p : Machine breaks down within a year. q : The company will replace it. Truth Table p q p q T T T T F F F T T F F T How do you express p q in terms of logical connectives? Method-1: Look at the true rows and take a OR.

23 Conditional statement p q If the machine breaks down within a year, the company will replace it. p : Machine breaks down within a year. q : The company will replace it. Truth Table p q p q T T T T F F F T T F F T How do you express p q in terms of logical connectives? Method-1: Look at the true rows and take a OR. Method-2: Look at the false rows, negate and take a AND.

24 Conditional statement p q If the machine breaks down within a year, the company will replace it. p : Machine breaks down within a year. q : The company will replace it. Truth Table p q p q T T T T F F F T T F F T How do you express p q in terms of logical connectives? Method-1: Look at the true rows and take a OR. Method-2: Look at the false rows, negate and take a AND. p q = p q.

25 Conditional statement p q If the machine breaks down within a year, the company will replace it. p : Machine breaks down within a year. q : The company will replace it.

26 Conditional statement p q If the machine breaks down within a year, the company will replace it. p : Machine breaks down within a year. q : The company will replace it. The company will replace the machine if it breaks down within a year.

27 Conditional statement p q If the machine breaks down within a year, the company will replace it. p : Machine breaks down within a year. q : The company will replace it. The company will replace the machine if it breaks down within a year. p q ; p is sufficient for q to happen.

28 Conditional statement p q If the machine breaks down within a year, the company will replace it. p : Machine breaks down within a year. q : The company will replace it. The company will replace the machine if it breaks down within a year. p q ; p is sufficient for q to happen. The company will replace the machine only if it breaks down within a year.

29 Conditional statement p q If the machine breaks down within a year, the company will replace it. p : Machine breaks down within a year. q : The company will replace it. The company will replace the machine if it breaks down within a year. p q ; p is sufficient for q to happen. The company will replace the machine only if it breaks down within a year. p q ; p is necessary for q to happen.

30 Conditional statement: some more examples p : You attend all lectures. q : You get an S grade. r : You score above 90 marks.

31 Conditional statement: some more examples p : You attend all lectures. q : You get an S grade. r : You score above 90 marks. 1. If you score above 90 marks you get an S grade. 2. Attending all lectures is not a sufficient condition for getting an S grade. 3. Attending all lectures is not a necessary condition for getting an S grade. 4. If you get an S grade then you attended all lectures or you scored above 90 marks. 5. You get an S grade only if you attend all lectures and you score above 90 marks.

32 Conditional statement: some more examples p : You attend all lectures. q : You get an S grade. r : You score above 90 marks. 1. If you score above 90 marks you get an S grade. r q 2. Attending all lectures is not a sufficient condition for getting an S grade. (p q) 3. Attending all lectures is not a necessary condition for getting an S grade. (q p) 4. If you get an S grade then you attended all lectures or you scored above 90 marks. q (p r) 5. You get an S grade only if you attend all lectures and you score above 90 marks. q (p r)

33 Conditional statement: examples revisited p : You attend all lectures. q : You get an S grade. r : You score above 90 marks. 1. Attending all lectures is not a sufficient condition for getting an S grade. (p q)

34 Conditional statement: examples revisited p : You attend all lectures. q : You get an S grade. r : You score above 90 marks. 1. Attending all lectures is not a sufficient condition for getting an S grade. (p q) You attend all lectures yet you do not get an S grade.

35 Conditional statement: examples revisited p : You attend all lectures. q : You get an S grade. r : You score above 90 marks. 1. Attending all lectures is not a sufficient condition for getting an S grade. (p q) You attend all lectures yet you do not get an S grade. 2. Attending all lectures is not a necessary condition for getting an S grade. (q p)

36 Conditional statement: examples revisited p : You attend all lectures. q : You get an S grade. r : You score above 90 marks. 1. Attending all lectures is not a sufficient condition for getting an S grade. (p q) You attend all lectures yet you do not get an S grade. 2. Attending all lectures is not a necessary condition for getting an S grade. (q p) You get an S grade but you need not have attended all lectures.

37 Conditional statement: examples revisited p : You attend all lectures. q : You get an S grade. r : You score above 90 marks. 1. Attending all lectures is not a sufficient condition for getting an S grade. (p q) You attend all lectures yet you do not get an S grade. 2. Attending all lectures is not a necessary condition for getting an S grade. (q p) You get an S grade but you need not have attended all lectures. Caution 1: Negation of a conditional statement is NOT a If.. then... statement. Caution 2: Necessary does NOT mean iff.

38 Conditional statement: mistaken with biconditional p : You attend all lectures. q : You get an S grade. r : You score above 90 marks. If you score above 90 marks you get an S grade. r q Incorrect implicit assumption: If you score below 90 marks you do not get an S grade.

39 Conditional statement: mistaken with biconditional p : A number is divisible by 6. q : A number is divisible by 3. p q If a number is divisible by 6 then it is divisible by 3. Clearly, does not mean than if a number is divisible by 3 then it is divisible by 6.

40 Biconditional statement p : A number is divisible by 6. q : A number is divisible by 3. r : A number is divisible by 2. p (q r) A number is divisible by 6 iff it is divisible by 2 and 3. p (q r) AND (q r) p.

41 Conditional statement: inverse, converse, contrapositive p : A number is divisible by 6. q : A number is divisible by 3. p q Converse: (q p) If a number is divisible by 3 then it is divisible by 6.

42 Conditional statement: inverse, converse, contrapositive p : A number is divisible by 6. q : A number is divisible by 3. p q Converse: (q p) If a number is divisible by 3 then it is divisible by 6. Inverse: ( p q) If a number is not divisible by 6 then it is not divisible by 3.

43 Conditional statement: inverse, converse, contrapositive p : A number is divisible by 6. q : A number is divisible by 3. p q Converse: (q p) If a number is divisible by 3 then it is divisible by 6. Inverse: ( p q) If a number is not divisible by 6 then it is not divisible by 3. Contrapositive: ( q p) If a number is not divisible by 3 then it is not divisible by 6.

44 Conditional statement: inverse, converse, contrapositive p : A number is divisible by 6. q : A number is divisible by 3. p q Converse: (q p) If a number is divisible by 3 then it is divisible by 6. Inverse: ( p q) If a number is not divisible by 6 then it is not divisible by 3. Contrapositive: ( q p) If a number is not divisible by 3 then it is not divisible by 6. Conditional statement equivalent to contrapositive.

45 Logical equivalences p q q p How do you prove it?

46 Logical equivalences p q q p How do you prove it? Truth Table p q p q q p T T T T T F F F F T T T F F T T

47 Logical equivalences p q q p How do you prove it? Truth Table p q p q q p T T T T T F F F F T T T F F T T Other methods? Intuition (risky!) Simplification using logical equivalences.

48 Logical equivalences Equivalence Name p T = Identity p T = Domination p p = Idempotent p q = q p Commutative (p q) r = p (q r) Associative p (q r) = (p q) (p r) Distributive (p q) = p q De Morgan s Law

49 Lets solve.. 1. Given that the value of p q is false, determine the value of (p q) q. 2. Write a compound statement that is true iff exactly two of the three statements p, q, r are true. 3. Show without truth table: (p q) = p q. 4. An island has two kinds of inhabitants the knaves who always lie and the knights who always tell the truth. You encounter two people A and B. A says: B is a knight. B says: The two of us are opposite types. What are A and B?

50 Limitations of propositions Recall why we defined propositions.. Using the following premises: If Milind has attended CS1100, he knows fundamentals of programming. Milind has attended CS1100. Conclusion: Milind knows fundamentals of programming.

51 Limitations of propositions Recall why we defined propositions.. Using the following premises: If Milind has attended CS1100, he knows fundamentals of programming. Milind has attended CS1100. Conclusion: Milind knows fundamentals of programming. However if we have these as our premises: Everyone who attends CS1100 knows fundamentals of programming. Milind has attended CS1100. How do we derive the conclusion?

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