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p ∨ q is logically equivalent to ________
- a)¬q → ¬p
- b)q → p
- c)¬p → ¬q
- d)¬p → q
Correct answer is option 'D'. Can you explain this answer?
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p ∨ q is logically equivalent to ________a)¬q → ¬pb)q ...
Understanding Logical Equivalences
In propositional logic, understanding how different logical expressions relate to each other is crucial. The expression "p ∨ q" (p or q) represents a disjunction, meaning at least one of p or q must be true.
Evaluating the Options
To determine which option is logically equivalent to p ∨ q, we can analyze each choice:
a) ¬q → ¬p
- This expression states that if q is false, then p must also be false.
- This does not represent the same truth values as p ∨ q.
b) q → p
- This means if q is true, then p is also true.
- Again, this does not capture the essence of p ∨ q.
c) ¬p → ¬q
- This implies if p is false, then q must also be false.
- This is not equivalent to p ∨ q.
d) ¬p → q
- This states that if p is false, then q must be true.
- This expression aligns with the truth table of p ∨ q. If p is false, q must be true for the disjunction to hold, making this the correct equivalent.
Truth Table Verification
Creating a truth table for p ∨ q and ¬p → q confirms:
- When both p and q are true: p ∨ q is true, ¬p → q is true.
- When p is true and q is false: p ∨ q is true, ¬p → q is true.
- When p is false and q is true: both p ∨ q and ¬p → q are true.
- When both p and q are false: p ∨ q is false, ¬p → q is false.
Hence, option d) ¬p → q is logically equivalent to p ∨ q.
In propositional logic, understanding how different logical expressions relate to each other is crucial. The expression "p ∨ q" (p or q) represents a disjunction, meaning at least one of p or q must be true.
Evaluating the Options
To determine which option is logically equivalent to p ∨ q, we can analyze each choice:
a) ¬q → ¬p
- This expression states that if q is false, then p must also be false.
- This does not represent the same truth values as p ∨ q.
b) q → p
- This means if q is true, then p is also true.
- Again, this does not capture the essence of p ∨ q.
c) ¬p → ¬q
- This implies if p is false, then q must also be false.
- This is not equivalent to p ∨ q.
d) ¬p → q
- This states that if p is false, then q must be true.
- This expression aligns with the truth table of p ∨ q. If p is false, q must be true for the disjunction to hold, making this the correct equivalent.
Truth Table Verification
Creating a truth table for p ∨ q and ¬p → q confirms:
- When both p and q are true: p ∨ q is true, ¬p → q is true.
- When p is true and q is false: p ∨ q is true, ¬p → q is true.
- When p is false and q is true: both p ∨ q and ¬p → q are true.
- When both p and q are false: p ∨ q is false, ¬p → q is false.
Hence, option d) ¬p → q is logically equivalent to p ∨ q.
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p ∨ q is logically equivalent to ________a)¬q → ¬pb)q ...
(p ∨ q) ↔ (¬p → q) is tautology.
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