6 Questions
What is the definition of logically equivalent statements?
Statements that have the same truth value in all possible scenarios
What is the truth value of the material implication 'if p, then q' when p is true and q is false?
False
What is the rule that states if p → q and p are true, then q must be true?
Rule of Detachment
What is the property that states if p → q and q → r are true, then p → r is true?
Transitive Property
What is the logical equivalence of ¬(p ∧ q)?
¬p ∨ ¬q
What is the logical equivalence of p ∧ (q ∨ r)?
(p ∧ q) ∨ (p ∧ r)
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)
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.
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