1. Logic and Truth Tables Winter 2012 COMP 1380 Discrete Structures I Computing Science Thompson Rivers University

2. TRU-COMP1380 Logic and Truth Tables2 Course Contents Speaking Mathematically –.5 weeks Number Systems and Computer Arithmetic – 2 weeks Logic and Truth Tables – 1 week Boolean Algebra and Logic Gates – 1 week Vectors and Matrices – 2 weeks Sets and Counting – 1.5 weeks Probability Theory and Distributions – 2 weeks Statics and Random Variables – 2 weeks

3. TRU-COMP1380 Logic and Truth Tables3 Unit Learning Objectives Recall truth tables for ~, , , , . Give a truth table for a given composite statement. Recall logical equivalences. Simplify a compound statement using logical equivalences. Infer a conclusion from a compound statement.

4. TRU-COMP1380 Logic and Truth Tables4 Unit Contents Sections 2.1 – 2.3 from the textbook Logical Form and Logical Equivalence Conditional Statements Valid and Invalid Arguments

5. Logical Form and Logical Equivalence Argument = premises + conclusion To have confidence in the conclusion in your argument, the premises should be acceptable on their own merits or follow from other statements that are known to be true. Logical forms for valid arguments? Examples Argument 1: If the program syntax is faulty or if program execution results in division by zero, then the computer will generate an error message. Therefore, if the computer does not generate, then the program syntax is correct and program execution does not result in division by zero. Argument 2: If x is a real number such that x 2, then x 2 > 4. Therefore, if x 2 /> 4, then x / 2. The common logical form of both of the above arguments: If p or q, then r. Therefore, if not r, then not p and not q. Is this logical form valid? TRU-COMP1380 Logic and Truth Tables5

6. Example If Jane is a major or Jane is a computing science major, then Jane will take COMP 1380. Jane is a computing science major. Therefore Jane will take COMP 1380. If logic is easy or (1), then (2). I will study hard. Therefore, I will get an A in this course. What are (1) and (2) in the above statement? TRU-COMP1380 Logic and Truth Tables6

7. Definition of statement A statement (or proposition) is a sentence that is true or false but not both. Examples Two plus two equals four. 2 + 2 = 4 I am a TRU student. x + y > 0??? TRU-COMP1380 Logic and Truth Tables7

8. Compound Statements Symbols used in complicated logical statements: ~not~pnegation of p  andp  qconjunction of p and q  orp  qdisjunction of p and q  exclusive orp  q Order of operations: ( ) and ~ have the precedence. ~p  q = (~p)  q ~(p  q) TRU-COMP1380 Logic and Truth Tables8

9. Example It is not hot but it is sunny.It is neither hot nor sunny. ->It is not hot, and it is sunny.It is not hot, and it is not sunny. Let h = “it is hot” and s = “it is sunny.” Then the above statements can be translated as ~h  s~h  ~s Example Suppose x is a particular real number. Let p, q, and r symbolize “0 < x,” “x < 3,” and “x = 3.” respectively. Then the following inequalities x  30 < x < 30 < x  3 can be translated as q  rp  qp  (q  r) TRU-COMP1380 Logic and Truth Tables9

10. Truth Tables Negation Conjunction Disjunction TRU-COMP1380 Logic and Truth Tables10 p~p~p T? F? pq p  q TT? TF? FT? FF? pq p  q TT? TF? FT? FF?

11. Truth Tables Exclusive or TRU-COMP1380 Logic and Truth Tables11 pq p  q TTF TFT FTT FFF

12. Example ~p  q ~T  F = ? ~T  T = ? (p  q)  ~(p  q) Can you write a truth table? When p is T and q is F? (p  q)  ~r When p is T, q is F and r is F? TRU-COMP1380 Logic and Truth Tables12 pq ~p  q TT? TF? FT? FF?

13. Logical Equivalence Example 6 > 2=2 < 6 How to prove p  q = q  pCommutative law p  q = q  p Commutative law ~(~p) = p Double negative law ~(p  q) ≠ ~p  ~q??? ~(p  q) ≠ ~p  ~q??? TRU-COMP1380 Logic and Truth Tables13

14. De Morgan’s Laws ~(p  q) = ~p  ~qCan you prove it? ~(statement1  statement2) = ~statement1  ~statement2 ~(p  q) = ~p  ~q ~(statement1  statement2) = ~statement1  ~statement2 Examples ~(~p  q) = ???~(p  ~q) = ??? The negation of “John is 6 feet tall and he weighs at least 200 pounds.” is... The negation of “The bus was late or Tom’s watch was slow.” is... The negation of “-1 < x  4” is... TRU-COMP1380 Logic and Truth Tables14

15. Tautologies and Contradictions p  ~p = T; p  T = T Always true p  ~p = F; p  F = F Always false Some other interesting equivalences p  F = pCan you prove it? p  T = p Associative Laws (p  q)  r = p  (q  r) (p  q)  r = p  (q  r) Distributive Laws p  (q  r) = (p  q)  (p  r)Can you prove it? p  (q  r) = (p  q)  (p  r) TRU-COMP1380 Logic and Truth Tables15

16. Prove ~(~p  q)  (p  q) = p. ~(~p  q)  (p  q) = (~(~p)  ~q)  (p  q) =... (p  ~q)  p = ??? ~((~p  q)  (~p  ~q) = ??? TRU-COMP1380 Logic and Truth Tables16

17. Conditional Statements Conditional Statement If hypothesis (or antecedent), then conclusion (or consequent). hypothesis  conclusion Example If 4686 is divisible by 6, then 4686 is divisible by 3. Is this statement True? Let p = “4686 is divisible by 6,” and q = “4686 is divisible by 3”. Then p  q Truth table for p  q p  q ≡ ~p  q ??? TRU-COMP1380 Logic and Truth Tables17 pq p  q TTT TFF FTT FFT

18. p  q  r≡ (p  r)  (q  r) ??? = ~(p  q)  r =... ~(p  q)≡ p  ~q ??? = ~(~p  q) =... p  q ≡ ~q  ~p ???Contrapositive = ~(~(p  q)) =... TRU-COMP1380 Logic and Truth Tables18

19. The biconditional of p and q p  q ≡ (p  q)  (q  p) p if and only if q p iff q The truth table is TRU-COMP1380 Logic and Truth Tables19 pq p  q TT? TF? FT? FF?

20. Valid and Invalid Arguments Example If (p  (q  ~r)) and (q  (p  r)), then is (p  r) valid? What do we have to do? p  r must be true for all the cases in which both of (p  (q  ~r)) and (q  (p  r)) are true. We may use the truth table to see if the statement is valid. Modus Ponents Both of p  q and p are valid, then q is valid. (p  q )  p = (~p  q )  p = (F  q )  T = q, or by truth table Therefore q must be T when (p  q ) and p are T. If x is a human, then x is mortal. Is it true? Dave is a human. Therefore Dave is mortal. TRU-COMP1380 Logic and Truth Tables20

21. Modus Tollens If p  q and ~q are valid, then ~p is valid. How to prove? If Zeus is human, then Zeus is mortal.Is it true? But Zeus is not mortal. Therefore... If x is divisible by 6, the x is divisible by 3. (Instantiation: …) 14 is not divisible by 3. Therefore... If a city is big, then the city has tall buildings. Because Kamloops have a tall building, Kamloops is a big city. Is this argument valid? TRU-COMP1380 Logic and Truth Tables21

22. Generalization p, then p  qHow to prove? Specialization p  q, then pHow to prove? Elimination p  q and ~q, then pHow to prove? Transitivity If p  q and q  r are valid, then p  r is valid.How to show? Contradiction Rule If p  F valid, then ~p is valid.How to show? The logical heart of the method of proof by contradiction. If an assumption leads to a contradiction, then that assumption must be false. TRU-COMP1380 Logic and Truth Tables22 pq p  q TTT TFF FTT FFT