*X*→ (¬*Z* →*Y)*

"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)

- $(X\land \neg Z) \rightarrow Y$
- $(X \land Y) \rightarrow \neg Z$
- $X \rightarrow(Y\land \neg Z)$
- $(X \rightarrow Y)\land \neg Z$

### 4 Comments

Basic Concept$:$ $ A$ `Unless`

$B \Leftrightarrow \neg B\rightarrow A$ `(Conditional Statement)`

$ \Leftrightarrow \neg(\neg B ) \vee A$

$ \Leftrightarrow B \vee A$

$ \Leftrightarrow A \vee B$

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- $P\rightarrow Q\equiv\neg P \vee Q$
- $Q$ Unless $\neg P\equiv P\rightarrow Q$

## 9 Answers

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:

- "If $P$ then $Q$"
- "$Q$ 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}$$

### 8 Comments

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?

@ShanthanK and @Lakshman Patel RJIT

I am interpreting the sentence as (__you cannot ride the roller coaster__ if __you are under 4 feet tall__) unless (__you are older than 16 years old__)

which gives (~q if r) unless (s)

which gives (r→ ~q) unless (s)

which gives (~s)→ (r→ ~q) (using p→ q: q unless ~p)

which gives (~s ^ r )→ ~q (using p→ (q→ r) = (p ^ q) → r )

which is the same as the ans given in Rosen book, i.e (r ^ ~s)→ ~q

**NOTE: **The only trick( that I feel) was in the last step

i.e, p→(q→r) = (p ^ q)→r

**This is also called as Exportation Law **. ( You can google it ;) )

If someone wants the proof of above, I am giving it here-

LHS: p→ (q→ r) = p→ (~q v r)

=~p v (~q v r)

=(~p v~q) v r

=~(p ^ q) v r

=(p ^ q)→ r : RHS

Correct me if I am wrong. Happy Learning guys.

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.

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

So option (a) is (X∧¬Z)→Y = ¬((X∧¬Z))∨Y = (¬X∨Z)∨Y, which matches our expression. So option A is correct.

## Option A is correct.

### 2nd interpretation....it is saying that if X happens then Y will happen otherwise Z will happen..So can we say that it means X and Z can not happen simultaneously because if X happens then output will be be Y not Z so can we also say that X happen and Z does not happen then output will be Y

## ( X ^ ~Z ) -> Y

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.

refer @varsha394 comment for knowing about exportation law. Now let's see its application in programming.

//(if X then Y) unless Z //(if X then Y) if not Z //(if X then Y) if ~Z //if ~Z then(if X then Y) if(Z is not true) { if(X is true) { print("Y is true") //Y is true } } //if X then (Y unless Z) //if X then (Y if not Z) //if X then (Y if ~Z) //if X then (if ~Z then Y) if(X is true) { if(Z is not true) { print("Y is true") //Y is true } } //By Exportation law,we can write //if X^~Z then Y // X^~Z -> Y if(X is true and Z is not true) { print("Y is true")//Y is true }