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"If X then Y unless Z" is represented by which of the following formulas in prepositional logic? ("$\neg$" is negation, "$\land$" is conjunction, and "$\rightarrow$" is implication)

  1. $(X\land \neg Z) \rightarrow Y$
  2. $(X \land Y) \rightarrow \neg Z$
  3. $X \rightarrow(Y\land \neg Z)$
  4. $(X \rightarrow Y)\land \neg Z$
asked in Mathematical Logic by Veteran (68.8k points)
retagged by | 1.8k views

X→ (¬ZY)

Basic Concept   A Unless B <=> ~B------->A
good trick sir ...thanks lakshman sir

5 Answers

+45 votes
Best answer
while ( not z )
{
if (X) then
    Y
}
or
unless( z ) 
{
if (X) then
    Y
}

this is what it means in programming. if you want to execute statement $Y$ then $X$ must be $\text{True}$ and $Z \text{False}$, which is equivalent to $(X\wedge \neg Z)Y.$

option A.

answered by Veteran (13.8k points)
edited by
Well explained
Someone please explain me this.
Cann't i write it as $xy + x'z$, if x is true then y else z
+24 votes
Answer is a) (X∧¬Z)→Y

((refer page 6,7 Discrete Math,ed 7,Kenneth H Rosen))

Implication "P implies Q" i.e (p→Q),where P is premise and Q is Conclusion, can be equivalently expressed in many ways. And the two equivalent expression relevant to the question are as follows

one is " if P then Q "

another is  "Q unless ¬P"  

both of these are equivalent to the propositional formula (P→Q),

Now compare "If X then Y unless Z" with  "Q unless ¬P" , here (¬P = Z) so (P = ¬Z) and (Q = Y)

compare with "if P then Q" , here (P = X) , (Q= Y)

So we get premise P= 'X' and '¬Z' ,conclusion Q = 'Y'

Equivalent propositional formula (X∧¬Z)→Y

EDIT:someone messaged me that i have taken "If X then (Y unless Z)" in above explanation and how to know if we take "(If X then Y) unless Z" or "If X then (Y unless Z)". So let me show both way gives same answer.

"(If X then Y) unless Z"  ⇔  "(X→Y)unless Z" ⇔ "¬Z→(X→Y)" ⇔ "¬Z→(¬X ⋁ Y)" ⇔ "Z ⋁ ¬X ⋁ Y"

 ⇔ "¬(X∧¬Z) ⋁ Y " ⇔ "(X∧¬Z)→Y"
answered by (321 points)
edited by
In Rosen, page:17,example:2, there is a similar question.

q:"you can ride roller coaster"
r:"you are under 4feet tall"
s:"you are older than 16"

For representing" you cannot ride the roller coaster if you are under 4 feet tall unless you are older than 16 years old"

(-q) if (r unless s)
(-q) if (-s --> r)
( -s --> r ) --> (-q) is the answer I"m getting, however, in the example he replaced unless with and not and gave answer (r ^ -s ) --> -q;

Now, which is correct?
Best explanation for this question.Thanks sourav.
Better Explanation ... Unless Can be treated as "OR"
+8 votes
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.
answered by Boss (7k points)
+1 vote
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.
answered by Boss (8.2k points)
+1 vote

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.

answered by Boss (8.2k points)
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