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Discrete Mathematics with Applications Chapter 2: The Logic of - - PowerPoint PPT Presentation

Mar 28, 2024 665 likes •1.62k views

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

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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

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Logical Equivalence Conditional Statements Conditional Equivalences

Two statement forms are called logically equivalent if, and

  • nly 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

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SLIDE 3

Logical Equivalence Conditional Statements Conditional Equivalences

Two statement forms are called logically equivalent if, and

  • nly 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

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Logical Equivalence Conditional Statements Conditional Equivalences

Two statement forms are called logically equivalent if, and

  • nly 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

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Logical Equivalence Conditional Statements Conditional Equivalences

Two statement forms are called logically equivalent if, and

  • nly 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

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Logical Equivalence Conditional Statements Conditional Equivalences

Two statement forms are called logically equivalent if, and

  • nly 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

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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

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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

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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

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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

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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

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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

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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

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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

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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

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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

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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

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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

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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

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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

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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

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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

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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

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Logical Equivalence Conditional Statements Conditional Equivalences

Appealing to the aforementioned common logical equivalences is helpful for verifying other, less obvious equivalences. e.g. Verify that ∼(∼p ∧ q) ∧ (p ∨ q) ≡ p.

Chapter 2: The Logic of Compound Statements (Part 2) Discrete Mathematics with Applications

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Logical Equivalence Conditional Statements Conditional Equivalences

Appealing to the aforementioned common logical equivalences is helpful for verifying other, less obvious equivalences. e.g. Verify that ∼(∼p ∧ q) ∧ (p ∨ q) ≡ p.

Chapter 2: The Logic of Compound Statements (Part 2) Discrete Mathematics with Applications

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Logical Equivalence Conditional Statements Conditional Equivalences

Appealing to the aforementioned common logical equivalences is helpful for verifying other, less obvious equivalences. e.g. Verify that ∼(∼p ∧ q) ∧ (p ∨ q) ≡ p. ∼(∼p ∧ q) ∧ (p ∨ q) ≡ (∼(∼p) ∨ ∼q) ∧ (p ∨ q) (De Morgan) ≡ (p ∨ ∼q) ∧ (p ∨ q) (Double Negative) ≡ p ∨ (∼q ∧ q) (Distributive Property) ≡ p ∨ c (Negation Law) ≡ p (Universal Bound Law)

Chapter 2: The Logic of Compound Statements (Part 2) Discrete Mathematics with Applications

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Logical Equivalence Conditional Statements Conditional Equivalences

If p and q are statement variables, the conditional of q by p is the statement “if p then q” or “p implies q.”

Chapter 2: The Logic of Compound Statements (Part 2) Discrete Mathematics with Applications

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Logical Equivalence Conditional Statements Conditional Equivalences

If p and q are statement variables, the conditional of q by p is the statement “if p then q” or “p implies q.” Notation: p → q.

Chapter 2: The Logic of Compound Statements (Part 2) Discrete Mathematics with Applications

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Logical Equivalence Conditional Statements Conditional Equivalences

If p and q are statement variables, the conditional of q by p is the statement “if p then q” or “p implies q.” Notation: p → q. p is called the hypothesis or antecedent.

Chapter 2: The Logic of Compound Statements (Part 2) Discrete Mathematics with Applications

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Logical Equivalence Conditional Statements Conditional Equivalences

If p and q are statement variables, the conditional of q by p is the statement “if p then q” or “p implies q.” Notation: p → q. p is called the hypothesis or antecedent. q is called the conclusion or consequent.

Chapter 2: The Logic of Compound Statements (Part 2) Discrete Mathematics with Applications

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Logical Equivalence Conditional Statements Conditional Equivalences

Example

p: “I am rich.” q: “I would buy a car.” p → q: “If I were rich, then I would buy a car.” (or “I would buy a car if I were rich.”)

Chapter 2: The Logic of Compound Statements (Part 2) Discrete Mathematics with Applications

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Logical Equivalence Conditional Statements Conditional Equivalences

The truth table for → p q p → q T T T T F F F T T F F T

Chapter 2: The Logic of Compound Statements (Part 2) Discrete Mathematics with Applications

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Logical Equivalence Conditional Statements Conditional Equivalences

The truth table for → p q p → q T T T T F F F T T F F T p → q is false only when p is true and q is false; otherwise it is true.

Chapter 2: The Logic of Compound Statements (Part 2) Discrete Mathematics with Applications

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Logical Equivalence Conditional Statements Conditional Equivalences

The truth table for → p q p → q T T T T F F F T T F F T p → q is false only when p is true and q is false; otherwise it is true. This should intuitively make sense. The only way that the statement “if p happens, then q also happens” could be false is if p happened, but q didn’t.

Chapter 2: The Logic of Compound Statements (Part 2) Discrete Mathematics with Applications

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Logical Equivalence Conditional Statements Conditional Equivalences

A conditional statement that is true because its hypothesis is false is called vacuously true or true by default.

Chapter 2: The Logic of Compound Statements (Part 2) Discrete Mathematics with Applications

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Logical Equivalence Conditional Statements Conditional Equivalences

A conditional statement that is true because its hypothesis is false is called vacuously true or true by default. Consider for instance the statement “If you show up for work Monday morning, then you will get the job.”

Chapter 2: The Logic of Compound Statements (Part 2) Discrete Mathematics with Applications

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Logical Equivalence Conditional Statements Conditional Equivalences

A conditional statement that is true because its hypothesis is false is called vacuously true or true by default. Consider for instance the statement “If you show up for work Monday morning, then you will get the job.” This statement is vacuously true if you do not show up for work.

Chapter 2: The Logic of Compound Statements (Part 2) Discrete Mathematics with Applications

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Logical Equivalence Conditional Statements Conditional Equivalences

A conditional statement that is true because its hypothesis is false is called vacuously true or true by default. Consider for instance the statement “If you show up for work Monday morning, then you will get the job.” This statement is vacuously true if you do not show up for work. This should make sense. The only way the above statement would be untrue would be if you do show up for work and yet do not get the job. (If you don’t show up to work, then the implication just doesn’t apply to you.)

Chapter 2: The Logic of Compound Statements (Part 2) Discrete Mathematics with Applications

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Logical Equivalence Conditional Statements Conditional Equivalences

Order of Operations

The precedence of the logical connectives is ∼ , then ∧, then ∨, and then → .

Chapter 2: The Logic of Compound Statements (Part 2) Discrete Mathematics with Applications

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Logical Equivalence Conditional Statements Conditional Equivalences

Order of Operations

The precedence of the logical connectives is ∼ , then ∧, then ∨, and then → . → associates to the left.

Chapter 2: The Logic of Compound Statements (Part 2) Discrete Mathematics with Applications

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Logical Equivalence Conditional Statements Conditional Equivalences

Order of Operations

The precedence of the logical connectives is ∼ , then ∧, then ∨, and then → . → associates to the left. But as always, parenthesizing complicated statement forms is always preferred.

Chapter 2: The Logic of Compound Statements (Part 2) Discrete Mathematics with Applications

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Logical Equivalence Conditional Statements Conditional Equivalences

Order of Operations

The precedence of the logical connectives is ∼ , then ∧, then ∨, and then → . → associates to the left. But as always, parenthesizing complicated statement forms is always preferred. e.g. p ∨ ∼q → r is (p ∨ ∼q) → r

Chapter 2: The Logic of Compound Statements (Part 2) Discrete Mathematics with Applications

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Logical Equivalence Conditional Statements Conditional Equivalences

Order of Operations

The precedence of the logical connectives is ∼ , then ∧, then ∨, and then → . → associates to the left. But as always, parenthesizing complicated statement forms is always preferred. e.g. p ∨ ∼q → r is (p ∨ ∼q) → r e.g. p → q ∧ ∼p → r is (p → (q ∧ ∼p)) → r

Chapter 2: The Logic of Compound Statements (Part 2) Discrete Mathematics with Applications

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Logical Equivalence Conditional Statements Conditional Equivalences

Example: The truth table for (p ∨ ∼q) → ∼p p q ∼q p ∨ ∼q ∼p (p ∨ ∼q) → ∼p T T F T F F T F T T F F F T F F T T F F T T T T

Chapter 2: The Logic of Compound Statements (Part 2) Discrete Mathematics with Applications

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Logical Equivalence Conditional Statements Conditional Equivalences

The “Division into Cases” Equivalence

Example: The truth table for (p ∨ q) → r ≡ (p → r) ∧ (q → r) p q r p ∨ q p → r q → r (p ∨ q) → r (p → r) ∧ (q → r) T T T T T T T T T T F T F F F F T F T T T T T T T F F T F T F F F T T T T T T T F T F T T F F F F F T F T T T T F F F F T T T T

Chapter 2: The Logic of Compound Statements (Part 2) Discrete Mathematics with Applications

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Logical Equivalence Conditional Statements Conditional Equivalences

Exercises

Suppose p, r are true and q is false. Find the truth value of the following statement forms.

Chapter 2: The Logic of Compound Statements (Part 2) Discrete Mathematics with Applications

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Logical Equivalence Conditional Statements Conditional Equivalences

Exercises

Suppose p, r are true and q is false. Find the truth value of the following statement forms.

1 ∼(p → q)

Chapter 2: The Logic of Compound Statements (Part 2) Discrete Mathematics with Applications

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Logical Equivalence Conditional Statements Conditional Equivalences

Exercises

Suppose p, r are true and q is false. Find the truth value of the following statement forms.

1 ∼(p → q) 2 ∼p → ∼q

Chapter 2: The Logic of Compound Statements (Part 2) Discrete Mathematics with Applications

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Logical Equivalence Conditional Statements Conditional Equivalences

Exercises

Suppose p, r are true and q is false. Find the truth value of the following statement forms.

1 ∼(p → q) 2 ∼p → ∼q 3 (p ∧ q) → r

Chapter 2: The Logic of Compound Statements (Part 2) Discrete Mathematics with Applications

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Logical Equivalence Conditional Statements Conditional Equivalences

Exercises

Suppose p, r are true and q is false. Find the truth value of the following statement forms.

1 ∼(p → q) 2 ∼p → ∼q 3 (p ∧ q) → r

Construct the truth tables for (p → q) → r and p → (q → r). Are these statement forms logically equivalent?

Chapter 2: The Logic of Compound Statements (Part 2) Discrete Mathematics with Applications

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Logical Equivalence Conditional Statements Conditional Equivalences

Representation of “If-Then” as “Or”

p → q ≡ ∼p ∨ q

Chapter 2: The Logic of Compound Statements (Part 2) Discrete Mathematics with Applications

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Logical Equivalence Conditional Statements Conditional Equivalences

Representation of “If-Then” as “Or”

p → q ≡ ∼p ∨ q

Chapter 2: The Logic of Compound Statements (Part 2) Discrete Mathematics with Applications

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Logical Equivalence Conditional Statements Conditional Equivalences

Representation of “If-Then” as “Or”

p → q ≡ ∼p ∨ q p q ∼p ∼p ∨ q p → q T T F T T T F F F F F T T T T F F T T T This is a SUPER IMPORTANT logical equivalence!!!

Chapter 2: The Logic of Compound Statements (Part 2) Discrete Mathematics with Applications

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Logical Equivalence Conditional Statements Conditional Equivalences

Example

Rewrite the statement “Either you get to work on time or you are fired.” in if-then form.

Chapter 2: The Logic of Compound Statements (Part 2) Discrete Mathematics with Applications

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Logical Equivalence Conditional Statements Conditional Equivalences

Example

Rewrite the statement “Either you get to work on time or you are fired.” in if-then form. Let ∼p be “You get to work on time.”

Chapter 2: The Logic of Compound Statements (Part 2) Discrete Mathematics with Applications

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Logical Equivalence Conditional Statements Conditional Equivalences

Example

Rewrite the statement “Either you get to work on time or you are fired.” in if-then form. Let ∼p be “You get to work on time.” Let q be “You are fired.”

Chapter 2: The Logic of Compound Statements (Part 2) Discrete Mathematics with Applications

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Logical Equivalence Conditional Statements Conditional Equivalences

Example

Rewrite the statement “Either you get to work on time or you are fired.” in if-then form. Let ∼p be “You get to work on time.” Let q be “You are fired.” Since ∼p ∨ q ≡ p → q, we can rewrite the statement as “If you do not get to work on time, then you are fired.”

Chapter 2: The Logic of Compound Statements (Part 2) Discrete Mathematics with Applications

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Logical Equivalence Conditional Statements Conditional Equivalences

Negation of a Conditional Statement

Using the fact that p → q ≡ ∼p ∨ q, we can see that ∼(p → q) ≡ ∼(∼p ∨ q) ≡ ∼(∼p) ∧ ∼q (De Morgan’s Law) ≡ p ∧ ∼q (Double Negative Law)

Chapter 2: The Logic of Compound Statements (Part 2) Discrete Mathematics with Applications

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Logical Equivalence Conditional Statements Conditional Equivalences

Negation of a Conditional Statement

Using the fact that p → q ≡ ∼p ∨ q, we can see that ∼(p → q) ≡ ∼(∼p ∨ q) ≡ ∼(∼p) ∧ ∼q (De Morgan’s Law) ≡ p ∧ ∼q (Double Negative Law) So the negation of “if p then q” is logically equivalent to “p and not q.”

Chapter 2: The Logic of Compound Statements (Part 2) Discrete Mathematics with Applications

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Logical Equivalence Conditional Statements Conditional Equivalences

Negation of a Conditional Statement

Using the fact that p → q ≡ ∼p ∨ q, we can see that ∼(p → q) ≡ ∼(∼p ∨ q) ≡ ∼(∼p) ∧ ∼q (De Morgan’s Law) ≡ p ∧ ∼q (Double Negative Law) So the negation of “if p then q” is logically equivalent to “p and not q.” This should make sense considering that p → q is false only when p is true and q is false.

Chapter 2: The Logic of Compound Statements (Part 2) Discrete Mathematics with Applications

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Logical Equivalence Conditional Statements Conditional Equivalences

Exercises

Negate each of the following statements:

Chapter 2: The Logic of Compound Statements (Part 2) Discrete Mathematics with Applications

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Logical Equivalence Conditional Statements Conditional Equivalences

Exercises

Negate each of the following statements:

1 “If my car is in the repair shop, then I cannot get to class.”

Chapter 2: The Logic of Compound Statements (Part 2) Discrete Mathematics with Applications

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Logical Equivalence Conditional Statements Conditional Equivalences

Exercises

Negate each of the following statements:

1 “If my car is in the repair shop, then I cannot get to class.” 2 “If Sara lives in Athens, then she lives in Greece.”

Chapter 2: The Logic of Compound Statements (Part 2) Discrete Mathematics with Applications

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Logical Equivalence Conditional Statements Conditional Equivalences

Exercises

Negate each of the following statements:

1 “If my car is in the repair shop, then I cannot get to class.” 2 “If Sara lives in Athens, then she lives in Greece.”

Possible Answers:

Chapter 2: The Logic of Compound Statements (Part 2) Discrete Mathematics with Applications

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Logical Equivalence Conditional Statements Conditional Equivalences

Exercises

Negate each of the following statements:

1 “If my car is in the repair shop, then I cannot get to class.” 2 “If Sara lives in Athens, then she lives in Greece.”

Possible Answers:

1 “My car is in the repair shop and I can get to class.” or “My

car is in the repair shop, but I can get to class.”

Chapter 2: The Logic of Compound Statements (Part 2) Discrete Mathematics with Applications

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Logical Equivalence Conditional Statements Conditional Equivalences

Exercises

Negate each of the following statements:

1 “If my car is in the repair shop, then I cannot get to class.” 2 “If Sara lives in Athens, then she lives in Greece.”

Possible Answers:

1 “My car is in the repair shop and I can get to class.” or “My

car is in the repair shop, but I can get to class.”

2 “Sara lives in Athens and she does not live in Greece.” or

“Sara lives in Athens, but she does not live in Greece.” (Sara might live in Georgia, or Ohio, or Wisconsin instead.)

Chapter 2: The Logic of Compound Statements (Part 2) Discrete Mathematics with Applications

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Logical Equivalence Conditional Statements Conditional Equivalences

Converse, Inverse, Contrapositive

For a conditional statement p → q

Chapter 2: The Logic of Compound Statements (Part 2) Discrete Mathematics with Applications

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Logical Equivalence Conditional Statements Conditional Equivalences

Converse, Inverse, Contrapositive

For a conditional statement p → q

1 its converse is q → p

Chapter 2: The Logic of Compound Statements (Part 2) Discrete Mathematics with Applications

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SLIDE 69

Logical Equivalence Conditional Statements Conditional Equivalences

Converse, Inverse, Contrapositive

For a conditional statement p → q

1 its converse is q → p 2 its inverse is ∼p → ∼q

Chapter 2: The Logic of Compound Statements (Part 2) Discrete Mathematics with Applications

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Logical Equivalence Conditional Statements Conditional Equivalences

Converse, Inverse, Contrapositive

For a conditional statement p → q

1 its converse is q → p 2 its inverse is ∼p → ∼q 3 its contrapositive is ∼q → ∼p

Chapter 2: The Logic of Compound Statements (Part 2) Discrete Mathematics with Applications

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Logical Equivalence Conditional Statements Conditional Equivalences

Converse, Inverse, Contrapositive

For a conditional statement p → q

1 its converse is q → p 2 its inverse is ∼p → ∼q 3 its contrapositive is ∼q → ∼p

Truth tables for each p q ∼p ∼q p → q q → p ∼p → ∼q ∼q → ∼p T T F F T T T T T F F T F T T F F T T F T F F T F F T T T T T T

Chapter 2: The Logic of Compound Statements (Part 2) Discrete Mathematics with Applications

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Logical Equivalence Conditional Statements Conditional Equivalences

Some Observations

A conditional statement p → q is logically equivalent to its contrapositive ∼q → ∼p.

Chapter 2: The Logic of Compound Statements (Part 2) Discrete Mathematics with Applications

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Logical Equivalence Conditional Statements Conditional Equivalences

Some Observations

A conditional statement p → q is logically equivalent to its contrapositive ∼q → ∼p. The converse q → p is logically equivalent to the inverse ∼p → ∼q.

Chapter 2: The Logic of Compound Statements (Part 2) Discrete Mathematics with Applications

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Logical Equivalence Conditional Statements Conditional Equivalences

Some Observations

A conditional statement p → q is logically equivalent to its contrapositive ∼q → ∼p. The converse q → p is logically equivalent to the inverse ∼p → ∼q. However, a conditional statement p → q is NOT logically equivalent to its converse q → p.

Chapter 2: The Logic of Compound Statements (Part 2) Discrete Mathematics with Applications

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Logical Equivalence Conditional Statements Conditional Equivalences

Example

Consider the statement “If it’s raining, then the ground is wet.” (Sounds reasonable)

Chapter 2: The Logic of Compound Statements (Part 2) Discrete Mathematics with Applications

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Logical Equivalence Conditional Statements Conditional Equivalences

Example

Consider the statement “If it’s raining, then the ground is wet.” (Sounds reasonable) Converse: “If the ground is wet, then it’s raining.” (This is a completely different statement. It’s also false.)

Chapter 2: The Logic of Compound Statements (Part 2) Discrete Mathematics with Applications

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Logical Equivalence Conditional Statements Conditional Equivalences

Example

Consider the statement “If it’s raining, then the ground is wet.” (Sounds reasonable) Converse: “If the ground is wet, then it’s raining.” (This is a completely different statement. It’s also false.) Contrapositive: “If the ground is not wet, then it’s not raining.” (This means the same thing as the original statement.)

Chapter 2: The Logic of Compound Statements (Part 2) Discrete Mathematics with Applications

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Logical Equivalence Conditional Statements Conditional Equivalences

Example

Consider the statement “If it’s raining, then the ground is wet.” (Sounds reasonable) Converse: “If the ground is wet, then it’s raining.” (This is a completely different statement. It’s also false.) Contrapositive: “If the ground is not wet, then it’s not raining.” (This means the same thing as the original statement.) Inverse: “If it’s not raining, then the ground is not wet.” (This means the same thing as the converse.)

Chapter 2: The Logic of Compound Statements (Part 2) Discrete Mathematics with Applications

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Logical Equivalence Conditional Statements Conditional Equivalences

Common English phrases for p → q or, equivalently, ∼q → ∼p:

Chapter 2: The Logic of Compound Statements (Part 2) Discrete Mathematics with Applications

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SLIDE 80

Logical Equivalence Conditional Statements Conditional Equivalences

Common English phrases for p → q or, equivalently, ∼q → ∼p:

1 If p, then q.

Chapter 2: The Logic of Compound Statements (Part 2) Discrete Mathematics with Applications

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SLIDE 81

Logical Equivalence Conditional Statements Conditional Equivalences

Common English phrases for p → q or, equivalently, ∼q → ∼p:

1 If p, then q. 2 p implies q.

Chapter 2: The Logic of Compound Statements (Part 2) Discrete Mathematics with Applications

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SLIDE 82

Logical Equivalence Conditional Statements Conditional Equivalences

Common English phrases for p → q or, equivalently, ∼q → ∼p:

1 If p, then q. 2 p implies q. 3 p only if q.

Chapter 2: The Logic of Compound Statements (Part 2) Discrete Mathematics with Applications

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SLIDE 83

Logical Equivalence Conditional Statements Conditional Equivalences

Common English phrases for p → q or, equivalently, ∼q → ∼p:

1 If p, then q. 2 p implies q. 3 p only if q. 4 p is a sufficient condition for q.

Chapter 2: The Logic of Compound Statements (Part 2) Discrete Mathematics with Applications

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SLIDE 84

Logical Equivalence Conditional Statements Conditional Equivalences

Common English phrases for p → q or, equivalently, ∼q → ∼p:

1 If p, then q. 2 p implies q. 3 p only if q. 4 p is a sufficient condition for q.

Common English phrases q → p or, equivalently, ∼p → ∼q:

Chapter 2: The Logic of Compound Statements (Part 2) Discrete Mathematics with Applications

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SLIDE 85

Logical Equivalence Conditional Statements Conditional Equivalences

Common English phrases for p → q or, equivalently, ∼q → ∼p:

1 If p, then q. 2 p implies q. 3 p only if q. 4 p is a sufficient condition for q.

Common English phrases q → p or, equivalently, ∼p → ∼q:

1 p if q

Chapter 2: The Logic of Compound Statements (Part 2) Discrete Mathematics with Applications

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SLIDE 86

Logical Equivalence Conditional Statements Conditional Equivalences

Common English phrases for p → q or, equivalently, ∼q → ∼p:

1 If p, then q. 2 p implies q. 3 p only if q. 4 p is a sufficient condition for q.

Common English phrases q → p or, equivalently, ∼p → ∼q:

1 p if q 2 p is a necessary condition for q

Chapter 2: The Logic of Compound Statements (Part 2) Discrete Mathematics with Applications

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SLIDE 87

Logical Equivalence Conditional Statements Conditional Equivalences

Common English phrases for p → q or, equivalently, ∼q → ∼p:

1 If p, then q. 2 p implies q. 3 p only if q. 4 p is a sufficient condition for q.

Common English phrases q → p or, equivalently, ∼p → ∼q:

1 p if q 2 p is a necessary condition for q

In other words, the conclusion of an implication expresses a necessary condition whereas the hypothesis expresses a sufficient condition.

Chapter 2: The Logic of Compound Statements (Part 2) Discrete Mathematics with Applications

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Logical Equivalence Conditional Statements Conditional Equivalences

Biconditional Statements

Given statement variables p and q, the biconditional of p and q is “p if and only if q” and is denoted by p ↔ q.

Chapter 2: The Logic of Compound Statements (Part 2) Discrete Mathematics with Applications

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Logical Equivalence Conditional Statements Conditional Equivalences

Biconditional Statements

Given statement variables p and q, the biconditional of p and q is “p if and only if q” and is denoted by p ↔ q. The words “if and only if” are sometimes abbreviated iff.

Chapter 2: The Logic of Compound Statements (Part 2) Discrete Mathematics with Applications

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SLIDE 90

Logical Equivalence Conditional Statements Conditional Equivalences

Biconditional Statements

Given statement variables p and q, the biconditional of p and q is “p if and only if q” and is denoted by p ↔ q. The words “if and only if” are sometimes abbreviated iff. Truth table for p ↔ q p q p ↔ q T T T T F F F T F F F T

Chapter 2: The Logic of Compound Statements (Part 2) Discrete Mathematics with Applications

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Logical Equivalence Conditional Statements Conditional Equivalences

Some Properties of the Biconditional

p ↔ q is true when p and q share the same truth values and is false when p and q have different truth values.

Chapter 2: The Logic of Compound Statements (Part 2) Discrete Mathematics with Applications

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SLIDE 92

Logical Equivalence Conditional Statements Conditional Equivalences

Some Properties of the Biconditional

p ↔ q is true when p and q share the same truth values and is false when p and q have different truth values. p ↔ q ≡ (p → q) ∧ (q → p)

Chapter 2: The Logic of Compound Statements (Part 2) Discrete Mathematics with Applications

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Logical Equivalence Conditional Statements Conditional Equivalences

Some Properties of the Biconditional

p ↔ q is true when p and q share the same truth values and is false when p and q have different truth values. p ↔ q ≡ (p → q) ∧ (q → p) A common English phrase for the biconditional statement p ↔ q is that “p is a necessary and sufficient condition for q.”

Chapter 2: The Logic of Compound Statements (Part 2) Discrete Mathematics with Applications