Logical Equivalence Conditional Statements Conditional Equivalences Discrete Mathematics with Applications Chapter 2: The Logic of Compound Statements (Part 2) January 25, 2019 Chapter 2: The Logic of Compound Statements (Part 2) Discrete Mathematics with Applications
Logical Equivalence Conditional Statements Conditional Equivalences Two statement forms are called logically equivalent if, and only if, they have identical truth values for each possible assignment of truth values to their statement variables. Chapter 2: The Logic of Compound Statements (Part 2) Discrete Mathematics with Applications
Logical Equivalence Conditional Statements Conditional Equivalences Two statement forms are called logically equivalent if, and only if, they have identical truth values for each possible assignment of truth values to their statement variables. Notation: If P and Q are logically equivalent, we write P ≡ Q . Chapter 2: The Logic of Compound Statements (Part 2) Discrete Mathematics with Applications
Logical Equivalence Conditional Statements Conditional Equivalences Two statement forms are called logically equivalent if, and only if, they have identical truth values for each possible assignment of truth values to their statement variables. Notation: If P and Q are logically equivalent, we write P ≡ Q . To test whether P and Q are logically equivalent: Chapter 2: The Logic of Compound Statements (Part 2) Discrete Mathematics with Applications
Logical Equivalence Conditional Statements Conditional Equivalences Two statement forms are called logically equivalent if, and only if, they have identical truth values for each possible assignment of truth values to their statement variables. Notation: If P and Q are logically equivalent, we write P ≡ Q . To test whether P and Q are logically equivalent: 1 Construct a truth table with one column for P and another column for Q . Chapter 2: The Logic of Compound Statements (Part 2) Discrete Mathematics with Applications
Logical Equivalence Conditional Statements Conditional Equivalences Two statement forms are called logically equivalent if, and only if, they have identical truth values for each possible assignment of truth values to their statement variables. Notation: If P and Q are logically equivalent, we write P ≡ Q . To test whether P and Q are logically equivalent: 1 Construct a truth table with one column for P and another column for Q . 2 Check for whether these two columns are identical. Chapter 2: The Logic of Compound Statements (Part 2) Discrete Mathematics with Applications
Logical Equivalence Conditional Statements Conditional Equivalences Example: double negation property: ∼ ( ∼ p ) ≡ p ∼ ( ∼ p ) p ∼ p T F T F T F Chapter 2: The Logic of Compound Statements (Part 2) Discrete Mathematics with Applications
Logical Equivalence Conditional Statements Conditional Equivalences Example: De Morgan’s Laws ∼ ( p ∧ q ) p q ∼ p ∼ q p ∧ q ∼ p ∨ ∼ q T T F F T F F T F F T F T T F T T F F T T F F T T F T T Chapter 2: The Logic of Compound Statements (Part 2) Discrete Mathematics with Applications
Logical Equivalence Conditional Statements Conditional Equivalences Example: De Morgan’s Laws ∼ ( p ∧ q ) p q ∼ p ∼ q p ∧ q ∼ p ∨ ∼ q T T F F T F F T F F T F T T F T T F F T T F F T T F T T So ∼ ( p ∧ q ) ≡ ∼ p ∨ ∼ q . Chapter 2: The Logic of Compound Statements (Part 2) Discrete Mathematics with Applications
Logical Equivalence Conditional Statements Conditional Equivalences Example: De Morgan’s Laws ∼ ( p ∧ q ) p q ∼ p ∼ q p ∧ q ∼ p ∨ ∼ q T T F F T F F T F F T F T T F T T F F T T F F T T F T T So ∼ ( p ∧ q ) ≡ ∼ p ∨ ∼ q . Exercise: use truth tables to show that ∼ ( p ∨ q ) ≡ ∼ p ∧ ∼ q . Chapter 2: The Logic of Compound Statements (Part 2) Discrete Mathematics with Applications
Logical Equivalence Conditional Statements Conditional Equivalences De Morgan’s Laws demystified Simply stated, the negation of an “and” statement is logically equivalent to the “or” statement in which each component is negated. Chapter 2: The Logic of Compound Statements (Part 2) Discrete Mathematics with Applications
Logical Equivalence Conditional Statements Conditional Equivalences De Morgan’s Laws demystified Simply stated, the negation of an “and” statement is logically equivalent to the “or” statement in which each component is negated. This should intuitively should make sense. In order for p ∧ q to be false, we would need p to be false or q to be false. Chapter 2: The Logic of Compound Statements (Part 2) Discrete Mathematics with Applications
Logical Equivalence Conditional Statements Conditional Equivalences De Morgan’s Laws demystified Simply stated, the negation of an “and” statement is logically equivalent to the “or” statement in which each component is negated. This should intuitively should make sense. In order for p ∧ q to be false, we would need p to be false or q to be false. The negation of an “or” statement is logically equivalent to the “and” statement in which each component is negated. Chapter 2: The Logic of Compound Statements (Part 2) Discrete Mathematics with Applications
Logical Equivalence Conditional Statements Conditional Equivalences De Morgan’s Laws demystified Simply stated, the negation of an “and” statement is logically equivalent to the “or” statement in which each component is negated. This should intuitively should make sense. In order for p ∧ q to be false, we would need p to be false or q to be false. The negation of an “or” statement is logically equivalent to the “and” statement in which each component is negated. Again, this should intuitively make sense. In order for p ∨ q to be false, both p and q need to be false. Chapter 2: The Logic of Compound Statements (Part 2) Discrete Mathematics with Applications
Logical Equivalence Conditional Statements Conditional Equivalences Write negations of each of the following statements in simple English. Chapter 2: The Logic of Compound Statements (Part 2) Discrete Mathematics with Applications
Logical Equivalence Conditional Statements Conditional Equivalences Write negations of each of the following statements in simple English. 1 John is 6 feet tall and he weighs at least 200 pounds. Chapter 2: The Logic of Compound Statements (Part 2) Discrete Mathematics with Applications
Logical Equivalence Conditional Statements Conditional Equivalences Write negations of each of the following statements in simple English. 1 John is 6 feet tall and he weighs at least 200 pounds. 2 The bus was late or Tom’s watch was slow. Chapter 2: The Logic of Compound Statements (Part 2) Discrete Mathematics with Applications
Logical Equivalence Conditional Statements Conditional Equivalences Write negations of each of the following statements in simple English. 1 John is 6 feet tall and he weighs at least 200 pounds. 2 The bus was late or Tom’s watch was slow. Answers: Chapter 2: The Logic of Compound Statements (Part 2) Discrete Mathematics with Applications
Logical Equivalence Conditional Statements Conditional Equivalences Write negations of each of the following statements in simple English. 1 John is 6 feet tall and he weighs at least 200 pounds. 2 The bus was late or Tom’s watch was slow. Answers: 1 John is not 6 feet tall or he weighs less than 200 pounds. Chapter 2: The Logic of Compound Statements (Part 2) Discrete Mathematics with Applications
Logical Equivalence Conditional Statements Conditional Equivalences Write negations of each of the following statements in simple English. 1 John is 6 feet tall and he weighs at least 200 pounds. 2 The bus was late or Tom’s watch was slow. Answers: 1 John is not 6 feet tall or he weighs less than 200 pounds. 2 The bus was not late and Tom’s watch was not slow. Chapter 2: The Logic of Compound Statements (Part 2) Discrete Mathematics with Applications
Logical Equivalence Conditional Statements Conditional Equivalences Write negations of each of the following statements in simple English. 1 John is 6 feet tall and he weighs at least 200 pounds. 2 The bus was late or Tom’s watch was slow. Answers: 1 John is not 6 feet tall or he weighs less than 200 pounds. 2 The bus was not late and Tom’s watch was not slow. WARNING: “The bus was early and Tom’s watch was fast” is an incorrect negation! Be careful with what the opposite of common English words are. Chapter 2: The Logic of Compound Statements (Part 2) Discrete Mathematics with Applications
Logical Equivalence Conditional Statements Conditional Equivalences Common Logical Equivalences Exercise: Verify these using truth tables. Chapter 2: The Logic of Compound Statements (Part 2) Discrete Mathematics with Applications
Logical Equivalence Conditional Statements Conditional Equivalences Appealing to the aforementioned common logical equivalences is helpful for verifying other, less obvious equivalences. Chapter 2: The Logic of Compound Statements (Part 2) Discrete Mathematics with Applications
Recommend
More recommend
