Propositional Logic Truth Tables Logical Equivalence Discrete Mathematics with Applications Chapter 2: The Logic of Compound Statements (Part 1) January 23, 2019 Chapter 2: The Logic of Compound Statements (Part 1) Discrete Mathematics with Applications
Propositional Logic Truth Tables Logical Equivalence Statements A statement (or proposition ) is a sentence that is either true (T) or false (F), but not both. Chapter 2: The Logic of Compound Statements (Part 1) Discrete Mathematics with Applications
Propositional Logic Truth Tables Logical Equivalence Statements A statement (or proposition ) is a sentence that is either true (T) or false (F), but not both. Examples: Chapter 2: The Logic of Compound Statements (Part 1) Discrete Mathematics with Applications
Propositional Logic Truth Tables Logical Equivalence Statements A statement (or proposition ) is a sentence that is either true (T) or false (F), but not both. Examples: 1 “The earth is flat.” – F Chapter 2: The Logic of Compound Statements (Part 1) Discrete Mathematics with Applications
Propositional Logic Truth Tables Logical Equivalence Statements A statement (or proposition ) is a sentence that is either true (T) or false (F), but not both. Examples: 1 “The earth is flat.” – F 2 “March has 31 days.” – T Chapter 2: The Logic of Compound Statements (Part 1) Discrete Mathematics with Applications
Propositional Logic Truth Tables Logical Equivalence Statements A statement (or proposition ) is a sentence that is either true (T) or false (F), but not both. Examples: 1 “The earth is flat.” – F 2 “March has 31 days.” – T 3 “Do you have the time?” – Not a statement (it’s a question) Chapter 2: The Logic of Compound Statements (Part 1) Discrete Mathematics with Applications
Propositional Logic Truth Tables Logical Equivalence Statements A statement (or proposition ) is a sentence that is either true (T) or false (F), but not both. Examples: 1 “The earth is flat.” – F 2 “March has 31 days.” – T 3 “Do you have the time?” – Not a statement (it’s a question) 4 “Go to the store.” – Not a statement (it’s a command) Chapter 2: The Logic of Compound Statements (Part 1) Discrete Mathematics with Applications
Propositional Logic Truth Tables Logical Equivalence Statements A statement (or proposition ) is a sentence that is either true (T) or false (F), but not both. Examples: 1 “The earth is flat.” – F 2 “March has 31 days.” – T 3 “Do you have the time?” – Not a statement (it’s a question) 4 “Go to the store.” – Not a statement (it’s a command) Notation: Lower case letters are often used to represent statements. Chapter 2: The Logic of Compound Statements (Part 1) Discrete Mathematics with Applications
Propositional Logic Truth Tables Logical Equivalence Logical Connectives Connectives are symbols that combine statements. Statements separated by connectives make a compound statement. Chapter 2: The Logic of Compound Statements (Part 1) Discrete Mathematics with Applications
Propositional Logic Truth Tables Logical Equivalence Logical Connectives Connectives are symbols that combine statements. Statements separated by connectives make a compound statement. The negation of p is the statement “not p ” or “it is not the case that p ” and is denoted by ∼ p . Chapter 2: The Logic of Compound Statements (Part 1) Discrete Mathematics with Applications
Propositional Logic Truth Tables Logical Equivalence Logical Connectives Connectives are symbols that combine statements. Statements separated by connectives make a compound statement. The negation of p is the statement “not p ” or “it is not the case that p ” and is denoted by ∼ p . The conjunction of the p and q is the statement “ p and q ” and is denoted by p ∧ q . Chapter 2: The Logic of Compound Statements (Part 1) Discrete Mathematics with Applications
Propositional Logic Truth Tables Logical Equivalence Logical Connectives Connectives are symbols that combine statements. Statements separated by connectives make a compound statement. The negation of p is the statement “not p ” or “it is not the case that p ” and is denoted by ∼ p . The conjunction of the p and q is the statement “ p and q ” and is denoted by p ∧ q . The disjunction of p and q is the statement “ p or q ” and is denoted by p ∨ q . Chapter 2: The Logic of Compound Statements (Part 1) Discrete Mathematics with Applications
Propositional Logic Truth Tables Logical Equivalence Inclusive Or vs. Exclusive Or NOTE: The meaning of “or” here is inclusive , that is p ∨ q is true whenever p is true, or q is true, or both. Chapter 2: The Logic of Compound Statements (Part 1) Discrete Mathematics with Applications
Propositional Logic Truth Tables Logical Equivalence Inclusive Or vs. Exclusive Or NOTE: The meaning of “or” here is inclusive , that is p ∨ q is true whenever p is true, or q is true, or both. e.g. “You will pass this course if you complete all of the homework or perform well on the exams.” Chapter 2: The Logic of Compound Statements (Part 1) Discrete Mathematics with Applications
Propositional Logic Truth Tables Logical Equivalence Inclusive Or vs. Exclusive Or NOTE: The meaning of “or” here is inclusive , that is p ∨ q is true whenever p is true, or q is true, or both. e.g. “You will pass this course if you complete all of the homework or perform well on the exams.” This is certainly a true statement, especially if you do both! Chapter 2: The Logic of Compound Statements (Part 1) Discrete Mathematics with Applications
Propositional Logic Truth Tables Logical Equivalence Inclusive Or vs. Exclusive Or NOTE: The meaning of “or” here is inclusive , that is p ∨ q is true whenever p is true, or q is true, or both. e.g. “You will pass this course if you complete all of the homework or perform well on the exams.” This is certainly a true statement, especially if you do both! When “or” is to be used in the exclusive sense, that is “ p or q but not both” or “ p or q but not both p and q , ” then we write p ⊕ q or p XOR q . Chapter 2: The Logic of Compound Statements (Part 1) Discrete Mathematics with Applications
Propositional Logic Truth Tables Logical Equivalence Examples p : “The earth is flat.” Chapter 2: The Logic of Compound Statements (Part 1) Discrete Mathematics with Applications
Propositional Logic Truth Tables Logical Equivalence Examples p : “The earth is flat.” q : “March has 31 days.” Chapter 2: The Logic of Compound Statements (Part 1) Discrete Mathematics with Applications
Propositional Logic Truth Tables Logical Equivalence Examples p : “The earth is flat.” q : “March has 31 days.” ∼ p : “The earth is not flat.” Chapter 2: The Logic of Compound Statements (Part 1) Discrete Mathematics with Applications
Propositional Logic Truth Tables Logical Equivalence Examples p : “The earth is flat.” q : “March has 31 days.” ∼ p : “The earth is not flat.” p ∧ q : “The earth is flat and March has 31 days.” Chapter 2: The Logic of Compound Statements (Part 1) Discrete Mathematics with Applications
Propositional Logic Truth Tables Logical Equivalence Examples p : “The earth is flat.” q : “March has 31 days.” ∼ p : “The earth is not flat.” p ∧ q : “The earth is flat and March has 31 days.” p ∨ q : “The earth is flat or March has 31 days.” Chapter 2: The Logic of Compound Statements (Part 1) Discrete Mathematics with Applications
Propositional Logic Truth Tables Logical Equivalence Order of Operations ∼ has the highest precedence, then ∧ , then ∨ . Chapter 2: The Logic of Compound Statements (Part 1) Discrete Mathematics with Applications
Propositional Logic Truth Tables Logical Equivalence Order of Operations ∼ has the highest precedence, then ∧ , then ∨ . ∧ and ∨ associate to the left – group two statements from the left. Chapter 2: The Logic of Compound Statements (Part 1) Discrete Mathematics with Applications
Propositional Logic Truth Tables Logical Equivalence Order of Operations ∼ has the highest precedence, then ∧ , then ∨ . ∧ and ∨ associate to the left – group two statements from the left. Parentheses can be used to indicate the order in which connectives should be taken. (It is a good habit to use parentheses to make compound statements more understandable.) Chapter 2: The Logic of Compound Statements (Part 1) Discrete Mathematics with Applications
Propositional Logic Truth Tables Logical Equivalence Order of Operations ∼ has the highest precedence, then ∧ , then ∨ . ∧ and ∨ associate to the left – group two statements from the left. Parentheses can be used to indicate the order in which connectives should be taken. (It is a good habit to use parentheses to make compound statements more understandable.) e.g. p ∧ ∼ q ∧ r means ( p ∧ ( ∼ q )) ∧ r Chapter 2: The Logic of Compound Statements (Part 1) Discrete Mathematics with Applications
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