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Logical Implication and Equivalence

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Questions and Answers

What is the definition of logically equivalent statements?

Statements that are never true

Statements that are always true

Statements that have different truth values

Statements that have the same truth value in all possible scenarios (correct)

What is the truth value of the material implication 'if p, then q' when p is true and q is false?

Either true or false

False (correct)

True

Undefined

What is the rule that states if p → q and p are true, then q must be true?

Rule of Detachment (correct)

De Morgan's Laws

Rule of Syllogism

Transitive Property

What is the property that states if p → q and q → r are true, then p → r is true?

Transitive Property (B) Signup and view all the answers

What is the logical equivalence of ¬(p ∧ q)?

¬p ∨ ¬q (A) Signup and view all the answers

What is the logical equivalence of p ∧ (q ∨ r)?

(p ∧ q) ∨ (p ∧ r) (C) Signup and view all the answers

Study Notes

Implication

Logical Equivalence

Definition: Two statements are said to be logically equivalent if they always have the same truth value (i.e., both are true or both are false) in all possible scenarios.

Material Implication

Definition: A material implication is a statement of the form "if p, then q" (p → q), which is false only when p is true and q is false.
Truth Table:
- p | q | p → q
- --- | --- | -----
- T | T | T
- T | F | F
- F | T | T
- F | F | T

Logical Equivalence of Implications

Rule of Detachment: If p → q and p are true, then q must be true.
Rule of Syllogism: If p → q and q → r are true, then p → r is true.
Transitive Property: If p → q and q → r are true, then p → r is true.

Logical Equivalence of Statements

De Morgan's Laws:
- ¬(p ∧ q) ≡ ¬p ∨ ¬q
- ¬(p ∨ q) ≡ ¬p ∧ ¬q
Distributive Property:
- p ∧ (q ∨ r) ≡ (p ∧ q) ∨ (p ∧ r)
- p ∨ (q ∧ r) ≡ (p ∨ q) ∧ (p ∨ r)

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Description

Test your understanding of logical implications, material implications, and logical equivalence of statements. Learn how to apply rules of detachment, syllogism, and transitive property to logical statements. Practice using De Morgan's Laws and the Distributive Property to simplify logical expressions.