COMP 1380 Discrete Structures I Thompson Rivers University Logic and Truth Tables COMP 1380 Discrete Structures I Computing Science Thompson Rivers University

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 Not the replacement of IPv6; Inter-transition mechanism; But if IPv6 would fail TRU-COMP1380 Logic and Truth Tables

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. TRU-COMP1380 Logic and Truth Tables

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

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 or x > 2, then x2 > 4. Therefore, if x2 /> 4, then x /< -2 and 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. [Q] Is the above logical form valid? TRU-COMP1380 Logic and Truth Tables

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 Tables

Compound Statements Symbols used in complicated logical statements: ~ not ~p negation of p  and p  q conjunction of p and q  or p  q disjunction of p and q  exclusive or p  q Order of operations: ( ) and ~ have the precedence. ~p  q = (~p)  q ~(p  q) TRU-COMP1380 Logic and Truth Tables

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 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  3 0 < x < 3 0 < x  3 can be translated as q  r p  q p  (q  r) TRU-COMP1380 Logic and Truth Tables

Truth Tables Negation Conjunction Disjunction p q p  q T ? F p q TRU-COMP1380 Logic and Truth Tables

Truth Tables Exclusive or p q p  q T F TRU-COMP1380 Logic and Truth Tables

Example ~p  q (p  q)  ~(p  q) (p  q)  ~r p q ~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? p q ~p  q T ? F TRU-COMP1380 Logic and Truth Tables

Logical Equivalence Example 6 > 2  2 < 6 How to prove 6 > 2  2 < 6 How to prove p  q  q  p Commutative 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 Tables

De Morgan’s Laws Examples ~(p  q) = ~p  ~q Can 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 Tables

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 = p Can 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 Tables

Prove ~(~p  q)  (p  q) = p using equivalences. ~(~p  q)  (p  q) = (~(~p)  ~q)  (p  q) = ... (p  ~q)  p = ??? Can you simplify using equivalences? ~((~p  q)  (~p  ~q)) = ??? TRU-COMP1380 Logic and Truth Tables

Conditional Statements 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 ??? p q p  q T F TRU-COMP1380 Logic and Truth Tables

p  q ≡ ~q  ~p ??? Contrapositive = ~(~(p  q)) 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 Tables

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 p q p  q T ? F TRU-COMP1380 Logic and Truth Tables

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. (It is like a rule.) Dave is a human. (It is like observation.) Therefore Dave is mortal. (It is like a conclusion.) TRU-COMP1380 Logic and Truth Tables

Modus Tollens If p  q and ~q are valid, then ~p is valid. How to prove? If x is human, then x is mortal. (rule) But Zeus is not mortal. (observation) Therefore ... (conclusion) If x is divisible by 6, the x is divisible by 3. (rule) (Instantiation: …) 14 is not divisible by 3. (fact) 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 Tables

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