“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) → Y
(B) (X ^ Y) → ¬ Z
(C) (X → (Y ^ ¬ Z)
(D) (X → Y(^ ¬ Z)


Answer: (A)

Explanation: The statement “If X then Y unless Z” means, if Z doesn’t occur, X implies Y i.e. ¬\negZ→\to(X→\toY), which is equivalent to Z∨\vee(X→\toY) (since P→\toQ ≡ ¬\negP∨\veeQ), which is then equivalent to Z∨\vee(¬\negX∨\veeY). Now we can look into options which one matches with this.

So option (a) is (X∧¬\wedge\negZ)→\toY = ¬\neg((X∧¬\wedge\negZ))∨\veeY = (¬\negX∨\veeZ)∨\veeY, which matches our expression. So option A is correct.

Source: http://www.cse.iitd.ac.in/~mittal/gate/gate_math_2002.html


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