1. CS104 The Foundations: Logic and Proof 1

2. 2 What is Discrete Structure?  Discrete Objects  Separated from each other (Opposite of continuous)  e.g., integers, people, house,  Vs. Continuous objects: e.g., real number  Discrete Structures  The abstract mathematical structures used to represent discrete objects and relationships between the objects e.g. sets, relations, graphs

3. 3 Why do we study Discrete Structures?  Information is stored and manipulated by computers in a discrete fashion. 0101101…  As a student in computer science major, you need to know the basic language and conceptual foundation for all of the computer science, i.e., Discrete Structures!  Discrete structure concepts are also widely used throughout math, science, engineering, economics, biology, etc., …  Get training for rational thought!

4. 1.1 Propositional Logic Introduction Propositions Propositional Logic Negation Connectives ( conjunction – disjunction – exclusive OR) Conditional statements (implication) Converse, Contrapositive and Inverse Biconditional Truth tables of Compound Propositions Precedence of Logic Operators Logic and Bit Operations 4

5. Introduction The rules of logic are used to distinguish between valid and invalid mathematical arguments. Logic has many applications to computer science. Ex: Computer circuits Construction of computer programs 5

6. Propositions The basic building blocks of logic. It is a declarative sentence (a sentence that declares a fact) that is either True or False, but not both. Example 1 1.Riyadh is the capital of Saudi Arabia. 2.Cairo is the capital of Lebanon. 3.1+1=2 4.2+2=3 Example 2 1.What time is it? 2.Read this carefully. 3.x+1=2 4.x+y=z All declarative sentences are propositions T – F – T – F Not propositions 1+2 not declarative sentences 3+4 neither true or nor false 6

7. Propositions We use letters to denote propositional variables that is used to represent propositions, just as letters used to denote numerical variables. The conventional letters used for propositions variables are p, q, r, s… The truth value if the proposition is true  T The truth value if the proposition is false  F The area of logic that deals with proposition is called the proposition calculus or propositional logic. 7

8. Propositions There are a methods for producing new propositions from those what we have. Many mathematical statement are constructed by combining one or more propositions (compound propositions) using logical operators. 8

9. Negation of a proposition Definition1 : Let p be a proposition. The negation of p, denoted by  p, is the statement “it is not the case that p”. The proposition  p is read “NOT p” Negation construct a new proposition from a single existing proposition. The truth table for the negation of a proposition. p pp FTFT TFTF Notice that  p is a proposition! 9

10. Negation of a proposition Example 3: Find the negation of the proposition “ Sara’s pc runs Linux ” and express this in simple English. Solution: Negation is: “ it is not the case that Sara’s pc runs Linux ” More simply: “ Sara’s pc does not runs Linux ” 10

11. Negation of a proposition Example 4: Find the negation of the proposition “ Ahmad’s smart phone has at least 32 GB of memory ” and express this in simple English. Solution: Negation is: “ Ahmad’s smart phone does not have at least32 GB of memory ” More simply: “ Ahmad’s smart phone has less than 32 GB of memory ” 11

12. Conjunction of propositions Definition2 : Let p and q be a proposition. The conjunction of p and q, denoted by p  q, is the proposition “p and q”. The conjunction p  q is true when both p and q are true, and is false otherwise. Sometimes “But” is used instead of “And” in conjunction. The truth table for the conjunction of two proposition. Notice that conjunction can be between two or more propositions pqp  q TTFFTTFF TFTFTFTF TFFFTFFF 12

13. Conjunction of propositions Example 5: Find the conjunction of the propositions p and q, where p is the proposition “ Ahmad’s pc has more than 16 GB free hard disk space ” and q is the proposition “ The processor in Ahmad’s pc runs faster than 1 GHz ” express this in simple English. Solution: Ahmad’s pc has more than 16 GB free hard disk space and The processor in Ahmad’s pc runs faster than 1 GHz. To be true, both conditions given must be true. It is false when one or both are false 13

14. Disjunction of propositions Definition3: Let p and q be a proposition. The disjunction of p and q, denoted by p  q, is the proposition “p or q”. The conjunction p  q is false when both p and q are false, and is true otherwise. The truth table for the disjunction of two proposition. Notice that disjunction can be between two or more propositions pqp  q TTFFTTFF TFTFTFTF TTTFTTTF 14

15. Disjunction of propositions Example 6: Find the disjunction of the propositions p and q, where p is the proposition “ Ahmad’s pc has more than 16 GB free hard disk space ” and q is the proposition “ The processor in Ahmad’s pc runs faster than 1 GHz ” express this in simple English. Solution: Ahmad’s pc has more than 16 GB free hard disk space or The processor in Ahmad’s pc runs faster than 1 GHz. When it will be true? False? 15

16. Exclusive-OR of propositions Definition4: Let p and q be a proposition. The exclusive or of p and q, denoted by p ⊕ q, is the proposition that is true when exactly one of p and q is true and is false otherwise. The truth table for the exclusive or of two proposition. Notice it can be between two or more propositions pqp ⊕ q TTFFTTFF TFTFTFTF FTTFFTTF 16

17. Note: inclusive or : The disjunction is true when at least one of the two propositions is true. E.g. “Students who have taken calculus or computer science can take this class.” – those who take one or both classes. exclusive or : The disjunction is true only when one of the proposition is true. E.g. “Students who have taken calculus or computer science, but not both, can take this class.” – only those who take one of them. 17

18. Conditional Statement (Implication) Definition5: Let p and q be a proposition. The conditional statement (implication) p  q is the proposition “if p, then q”. The conditional statement p  q is false when p is true and q is false, otherwise true. In the conditional statement p  q, p is called the hypothesis or (antecedent or premise) and q is called the conclusion or (consequence). Comes in format: If p, then q If p, q p is sufficient for q p implies q … you will find rest of them in book. Look at them! 18

19. Conditional Statement (Implication) The truth table for the implication of two proposition. Ex: “If you get 100% on the final, then you will get an A”. pqp  q TTFFTTFF TFTFTFTF TFTTTFTT HypothesisConclusion 19

20. Conditional Statement (Implication) Example If Juan has a smart phone, then 2+3 = 5 This proposition is true, because conclusion is true. If Juan has a smart phone, then 2+3 = 6 This proposition is true, if Juan does not have a smart phone even though 2+3=6 is false. There are no relation between hypothesis and conclusion. In programming we have if-then. HypothesisConclusion 20

21. Conditional Statement (Implication) Example 7: Let p the statement “ Maria learns discrete mathematics ” and q the statement “ Maria will find job ” express the statement p  q as a statement in English. Solution: If Maria learns discrete mathematics, then she will find job. Maria will find job when she learns discrete mathematics. Rest example in book page 7 When it will be true? False? 21

22. Converse, Contrapositive and Inverse  Converse: q  p is converse of p  q. Ex.: p  q: “If it is noon, then I am hungry.” q  p: “If I am hungry, then it is noon.”  Contrapositive:  q   p is contapositive of p  q. Ex.:  q   p: “If I am not hungry, then it is not noon.”  Inverse:  p   q is inverse of p  q. Ex.:  p   q: “If it is not noon, then I am not hungry.” p  q has same truth values as  q   p See example 7 22

23. Biconditional Statement Definition6: Let p and q be two propositions. The biconditional statement (bi-implication) p ↔ q is the proposition “p if and only if q”. The biconditional statement p ↔ q is true when p and q have the same truth values, otherwise false. It corresponds to English “p if and only if q”. The truth table of biconditional pqp ↔ q TTFFTTFF TFTFTFTF TFFTTFFT 23

24. Biconditional Statement Example10: p: “You can take the flight” q: “You buy a ticket” p ↔ q: “You can take the flight if and only if you buy a ticket” True if p and q both true or both false False otherwise 24

25. Implicit Biconditional Statement Not always explicit in natural language. Often expressed using an “if, then” or “only, if” Converse is implied but not stated. Ex: “if you finish your meal, then you can have dessert” Means “You can have dessert if and only if you finish your meal” 25

26. Truth Table of Compound Proposition Used to build up complicated compound positions involving any numbers of propositional variables. Example: Construct the truth table of the compound proposition (p ν ¬q) → (p Λ q). The Truth Table of (p ν ¬q) → (p Λ q). p q¬q¬qp ν ¬qp Λ qp Λ q (p ν ¬q) → (p Λ q) T T F F T F FTFTFTFT TTFTTTFT TFFFTFFF TFTFTFTF 26  Q3: Construct truth table for  (p   q) ↔ (p  q).  (p ⊕  q) → (p ⊕ q). # of raws = 2 # of variable

27. Precedence 27  Precedence of Logical Operators: OperatorPrecedence ( )¬Λν→↔ ( )¬Λν→↔ 123456123456

28. Logic and Bit Operations 28  Bit: A bit is a symbol with two possible values, namely, 0 (zero) and 1 (one).  A bit can be used to represent a truth value as 1 for T and 0 for F  Bit string: A bit string is a sequence of bits. The length of the string is number of bits in the string.  Example: 10101001 is a bit string of length eight  We define the bitwise OR, AND, and XOR of two strings of same length to be the strings that have as their bits the OR, AND, and XOR of the corresponding bits in the two strings, respectively.  We use the symbols , , and ⊕ to represent bitwise OR, AND, and XOR, respectively.

29. Logic and Bit Operations x ⊕ y x  yx  y yx 01100110 00010001 01110111 01010101 00110011 29 Truth table for bitwise OR, AND, and XOR: Example 13 Find bitwise OR, AND, and XOR of the bit strings 0110110110 and 1100011101