GATE CSE 2002 | Question: 1.8

Question

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

• A. $(X\land \neg Z) \rightarrow Y$
• B. $(X \land Y) \rightarrow \neg Z$
• C. $X \rightarrow(Y\land \neg Z)$
• D. $(X \rightarrow Y)\land \neg Z$

Comments

good trick sir ...thanks lakshman sir

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it helped me thanks..!

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$A \text{  }\underline{\text{unless}}\text{  } B \equiv A + B$

$\therefore ($If $X$ then $Y)$ unless $Z \equiv ($If $X$ then $Y) + Z \equiv X’  + Y + Z$

$\equiv$ If $X$ then $(Y$ unless $Z) \equiv $If $X$ then $(Y + Z) \equiv X’  + Y + Z$

 

Option A: $(X \land \neg Z) \rightarrow Y = (XZ')' + Y = X' + Z + Y =$ (Ans.)

Option B: $(X \land Y) \rightarrow \neg Z = X' + Y' + Z'$

Option C: $X \rightarrow (Y \land \neg Z) = X' + YZ'$

Option D: $(X \rightarrow Y) \land \neg Z = (X' + Y)Z' = X'Z' + YZ'$

Answer 1

Answer is a) $(X \land \lnot Z)\to Y$

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

Implication "$P$ implies $Q$" i.e., $(p \to Q)$, 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:

1. "If $P$ then $Q$"
2. "$Q$ unless $\lnot P$" 

Both of these are equivalent to the propositional formula $(P \to Q)$,

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

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

So we get premise $P= X \text{ and } \lnot Z,$ conclusion $Q = Y$

Equivalent propositional formula $(X \land \lnot Z) \to Y$

PS: 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 that both way gives the same answer.

"(If $X$ then $Y$) unless $Z$"  $\equiv (X\to Y)$ unless $Z$

$$\begin{align}
&\equiv \lnot Z \to (X\to Y) \\
&\equiv \lnot Z\to (\lnot X \lor Y) \\
&\equiv Z \lor \lnot X \lor Y \\
&\equiv \lnot (X \land \lnot Z) \lor Y \\
&\equiv (X \land \lnot Z) \to Y
\end{align}$$

Comments

Thanks for letting me know the exportation law.

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@Arjun @Deepak Poonia sir,

 

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:

1. "If P then Q"
2. "Q unless ¬P” 

Sir here I think point 2 is wrong bcz the contrapositive statement of (P→ Q) will be (~Q → ~P). So we can say that ~P unless Q, but in the answer reverse is written.

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In Propositional Logic:

$Unless = If \,\,\, \, Not$

$Provided \,\,\, That = If$

So, $\text{A unless B} == \text{A If Not B} == \text{A If  } \neg B == \neg B \rightarrow A == B \vee A $

And $\text{B unless A} == \text{B  If Not A} == \neg A \rightarrow B == A \vee B == \text{A unless B} $

So, ~P unless Q is same as Q unless ~P.

Watch @GO Classes Discrete mathematics 2023 course, Propositional Logic, Lecture 11, which covers “Converse, English-Logic Translation, Unless” etc. Watch this lecture. It will help you understand many concepts very clearly.

https://www.goclasses.in/s/store/courses/description/2023-Discrete-Mathematics

Answer 2

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).

Comments

your code is not quite correct

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Above code is a pseudo code → It is correct.

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@aryavart

if, unless indicates conditional statements

while, until indicates loop statements

 

Answer 3

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.

Comments

Z∨ ( ¬X∨ Y) =  (¬X∨ Z)∨ Y because OR is associative.

Answer 4

I just to tried to simplify it more

Statement : IF X THEN Y UNLESS Z

we can write it like this (IF X THEN Y) UNLESS Z

whenever you see UNLESS  operator just keep in mind to replace it with "IF NOT " your work is half done

=> (IF X THEN Y) IF NOT Z

=> (IF X THEN Y) IF (NOT Z)

now see it falls in the form of Q if P which is another form of P->Q

=> ~Z -> (IF X THEN Y )

=> ~Z -> (X -> Y)

=> ~Z -> ( ~X ∨ Y)

=>~(~Z) ∨ ( ~X ∨ Y)

=> Z ∨ ~X ∨ Y

=> (Z ∨ ~X ) ∨ Y

=> ~ (~Z ∧ X ) ∨ Y

=>  (~Z ∧ X ) -> Y