CMSC 250 Syllabus Lecture Section TuTh 12 30 1 45 Lab Discussion Section M W 10 10 50 or 11 11 50 Every Week Worksheet Quiz Homework 2 Hourly Exams Final as noted on Syllabus Web Page Motivation Why Learn This Material Some things can be directly applied Some things are good to know Some things just teach a way of thinking and expressing yourself Overall Theme Proofs 1 Course Content Propositional Logic and circuits Predicate Calculus quantification Number Theory Mathematical Induction Counting Combinations and Probability Functions Relations Graph Theory Statement Proposition declarative makes a statement can be understood to be either true or false in an interpretation symbolized by a letter Examples Today is Wednesday 5 2 7 3 6 18 The sky is blue Why is the sky blue Brett Favre This sentence is false 2 Other Symbols and Definitions to make compound statements Conjunction and symbolized by Disjunction or symbolized by Negation not symbolized by Truth Tables for these operators Translation of English to Symbolic Logic Statements The sky is blue one simple atomic statement assign to a letter i e b The sky is blue and the grass is green one statement conjunction of two atomic statements each single statement gets a letter i e b g and join with i e b g The sky is blue or the sky is purple one statement disjunction of two atomic statements each single statement gets a letter i e b p and join with i e b p 3 Trickier Translation 1 The sky is blue or purple two statements the sky is blue assign this to b the sky is purple assign this to p still a disjunction the sky is blue or the sky is purple bvp Trickier Translation2 The sky is blue but not dark two statements the sky is blue assign this to b the sky is dark assign this to d conjunction with negation the sky is blue and the sky is not dark the sky is blue and it is not the case that the sky is dark it is not the case that the sky is dark b d is d 4 Trickier Translation 3 2 x 6 English x is greater than or equal to 2 and less than or equal to 6 two statements x is greater than or equal to 2 assign this to p x is less than or equal to 6 assign this to q becomes p q 3 Continued 2 x 6 p is actually a compound statement r s x is greater than 2 is symbolized by r x is equal to 2 is symbolized by s q is actually a compound statement x is less than 6 or x is equal to 6 m n x is less than 6 is symbolized by m x is equal to 6 is symbolized by n p q becomes r s m n x is greater than 2 or x is equal to 2 5 More about Operators exclusive or p q p or q but not both p q same as p q p q Precedence between the operators not highest precedence and or have equal precedence use parentheses to override default precedence a b c Other More Advanced Truth Tables p r q r p r p r p q q p 6 Special Results in the Truth Table Tautological Proposition a tautology is a statement that can never be false when all of the lines of the truth table have the result true Contradictory Proposition a contradiction is a statement that can never be true when all of the lines of the truth table have the result false Logical Equivalence of two propositions two statements are logically equivalent if they will be true in exactly the same cases and false in exactly the same cases when all of the lines of one column of the truth table have all of the same truth values as the corresponding lines from another column of the truth table Special Truth Tables p p p p p r p r p q q p p q q p 7 Logical Equivalences Theorem 1 1 1 Page 14 Double Negative p p Commutative p q q p and p q q p Associative p q r p q r and p q r p q r Distributive p q r p q p r and p q r p q p r More Logical Equivalences Idempotent p p p and p p p Absorption p p q p and p p q p Identity p t p and p c p Negation p p t and p p c Universal Bound p c c and p t t Negations of t and c t c and c t 8 DeMorgan s Laws p q p q p q p q It is not the case that Pete or Quincy went to the store Pete did not go to the store and Quincy did not go to the store It is not the case that both Pete and Quincy went to the store Pete did not go to the store or Quincy did not go to the store Prove by Truth Table by Rules p q q p p p q p q p q p p q p q p q q p 9 Conditional Statements Hypothesis Conclusion If this then that Hypothesis implies Conclusion has lowest precedence If it is raining I will carry my umbrella If you don t eat your dinner you will not get desert p q 1 1 0 0 1 0 1 0 p q 1 0 1 1 Converting to p q p q Show with Truth Table p q p q Show with Truth Table and Rules 10 Contrapositive Negate the conclusion and negate the hypothesis Use the negated Conclusion as the new Hypothesis and the negated Hypothesis as the Conclusion p q q p English If Paula is here then Quincy is here If Quincy is not here then Paula is not here Converse and Inverse p q If Paula is here then Quincy is here Converse swap the hypothesis and the conclusion q p If Quincy is here then Paula is here Inverse negate the hypothesis and negate the conclusion p q If Paula is not here then Quincy is not here 11 biconditional p if and only if q p q p iff q p q p q 1 1 0 0 1 0 1 0 1 0 0 1 Only If Translation to if then form p only if q p can be true only if q is true if q is not true then p can t be true if not q then not p q p if p then q p q Translation in English You will graduate is CS only if you pass this course G only if P If you do not pass this course then you will not graduate in CS P G If you graduate in CS then you passed this course G P 12 Other English Words for Implication Sufficient Condition if r then s r s means …
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