Propositional Logic: Definitions, Connectives, and Examples
Explore the fundamentals of propositional logic including propositions, connectives, compound propositions, and their relationships. Learn about logical operators like AND, OR, IF...THEN, and NOT to form complex logical statements. Dive into examples to grasp the concepts effectively.
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CS2209 Tutorial 1 Sep. 14th , 2020 TA: Xindi Wang xwang842@uwo.ca Department of Computer Science
Reminder Assignment 1 is due on Sep 20th in zyBook (which means that you are required to buy zyBook to finish the assignment) Presentation Title Here
Todays Agenda Propositions and Connectives Propositional Logic Equivalence Presentation Title Here
Propositions and Connectives Propositions and Connectives
Propositions Definitions: A proposition is a statement that is either True or False Two plus two is four/ two plus two is zero Questions, commands, requests are not propositions Propositional variables: is a variable which can either be true or false X, Y, Z (or p, q, r) Propositions
Connectives Definitions: A logical connective is a symbol which is used to connect one or more propositions. Generally there are four connectives: conjunction(AND), disjunction(OR), implication(IF...THEN...), negation(NOT) Three conditional statements related to implication(? ?): converse (q p), contrapositive( q p), inverse ( p q) Connectives
Connectives Notation False when ? ? p AND q Either p or q must be false ? ? p OR q Both p and q must be false ? ? IF p, THEN q If p is true, then b is false ? NOT p p is true ? ? p XOR q p and q have the same truth value Connectives
Compound propositions Definitions: consists of two or more propositions joined by connectives Precedence of the connectives: first, then , then , last. (use parentheses when in doubt or need a different order) A B C A is equal to ((A ( B)) ( C)) A Note: A B C is not equal to (A B) C Compound Propositions
Compound propositions Example: B C A A B C B C A A B C T T T F T F T T F F F T T F T F F T T F F F F T F T T T T T F T F T F T F F T T F T F F F T F T Compound Propositions
Propositional Logic Equivalence Propositional Logic Equivalence
Equivalence Definitions: Two compound propositions F and G are logically equivalent (? ?) if they have the same value for every row in the truth table on their variables. ( ? ?) (? ?) ? Proving that ? ? can be done by proving that ? ? is tautology (tautology: the proposition is always true, contradiction: the proposition is always false) Equivalence
Equivalence ? ? A A B A B T T F T T F F F F T T T F F T T ? ?) A B A B T T T T F F F T T F F T Equivalence
Equivalence Double negation: ? ? De Morgan s Laws (? ?) ( ? ?) (? ?) ( ? ?) Equivalence
Equivalence Example: use truth tables to prove logical equivalence ? ? and ? ? (p p p p q p q q) q T T T F F F T F F T F T F T F T T T F F T F T F Equivalence
