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“If X, then Y unless Z” is represented by which of the following formulae in propositional logic?
(“¬” is negation “^” is conjunction, and “→” is implication)
(“¬” is negation “^” is conjunction, and “→” is implication)
- a)(X ^ ¬ Z) → Y
- b)(X ^ Y) → ¬ Z
- c)(X → (Y ^ ¬ Z)
- d)(X → Y(^ ¬ Z)
Correct answer is option 'A'. Can you explain this answer?
Most Upvoted Answer
“If X, then Y unless Z” is represented by which of the fol...
Understanding the Statement
The statement "If X, then Y unless Z" can be broken down into its components:
- X: A condition or premise.
- Y: The conclusion that follows if X is true.
- Z: An exception that negates the conclusion Y.
Logical Interpretation
The phrase "unless Z" indicates that Y will only be true if Z is not true. Therefore, the statement can be interpreted as:
- If X is true and Z is not true, then Y must be true.
This leads us to the logical structure of the statement.
Breaking Down the Options
Let's analyze the options provided:
- a) (X ^ ¬ Z) → Y: This translates to "If X is true and Z is false, then Y is true." This directly aligns with our interpretation of the original statement.
- b) (X ^ Y) → ¬ Z: This suggests that if both X and Y are true, then Z must be false, which does not capture the original meaning.
- c) (X → (Y ^ ¬ Z): This implies that if X is true, then both Y is true and Z is false, which is a stronger statement than intended.
- d) (X → Y(^ ¬ Z): This option is ambiguous and does not clearly represent the original statement.
Conclusion
The correct representation of "If X, then Y unless Z" is option a) (X ^ ¬ Z) → Y, as it accurately encapsulates the conditional relationship and the exception presented in the statement.
The statement "If X, then Y unless Z" can be broken down into its components:
- X: A condition or premise.
- Y: The conclusion that follows if X is true.
- Z: An exception that negates the conclusion Y.
Logical Interpretation
The phrase "unless Z" indicates that Y will only be true if Z is not true. Therefore, the statement can be interpreted as:
- If X is true and Z is not true, then Y must be true.
This leads us to the logical structure of the statement.
Breaking Down the Options
Let's analyze the options provided:
- a) (X ^ ¬ Z) → Y: This translates to "If X is true and Z is false, then Y is true." This directly aligns with our interpretation of the original statement.
- b) (X ^ Y) → ¬ Z: This suggests that if both X and Y are true, then Z must be false, which does not capture the original meaning.
- c) (X → (Y ^ ¬ Z): This implies that if X is true, then both Y is true and Z is false, which is a stronger statement than intended.
- d) (X → Y(^ ¬ Z): This option is ambiguous and does not clearly represent the original statement.
Conclusion
The correct representation of "If X, then Y unless Z" is option a) (X ^ ¬ Z) → Y, as it accurately encapsulates the conditional relationship and the exception presented in the statement.
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“If X, then Y unless Z” is represented by which of the fol...
The statement “If X then Y unless Z” means, if Z doesn’t occur, X implies Y i.e. ¬Z→(X→Y), which is equivalent to Z ∨ (X→Y)
(since P→Q ≡ ¬P ∨ Q), which is then equivalent to Z ∨ (¬X ∨ Y). Now we can look into options which one matches with this.
So option (a) is (X∧¬Z)→Y = ¬( (X∧¬Z) ) ∨ Y = (¬X∨Z) ∨ Y, which matches our expression. So option (A) is correct.
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“If X, then Y unless Z” is represented by which of the following formulae in propositional logic?(“¬” is negation “^” is conjunction, and “→” is implication)a)(X ^ ¬ Z) → Yb)(X ^ Y) → ¬ Zc)(X → (Y ^ ¬ Z)d)(X → Y(^ ¬ Z)Correct answer is option 'A'. Can you explain this answer?
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