TIFR CSE 2010 | Part A | Question: 4

Question

• If the bank receipt is forged, then Mr. M is liable.
• If Mr. M is liable, he will go bankrupt.
• If the bank will loan him money, he will not go bankrupt.
• The bank will loan him money.

Which of the following can be concluded from the above statements?

• A. Mr. M is liable
• B. The receipt is not forged
• C. Mr. M will go bankrupt
• D. The bank will go bankrupt
• E. None of the above

Answer 1

Let us denote sentences with variables:

• $F:$ Bank receipt is forged
• $L:$ Mr. $M$ is liable
• $B:$ He will go bankrupt
• $M:$ Bank loan him money

1. $F\to L \quad \to(1)$
2. $L\to B \quad \to(2)$
3. $M\to B'\quad \to(3)$
4. $M\quad\qquad \to(4)$

From $3$ and $4$ modus ponens we get

• $B' \quad \to(5)$

From $2$ and $5$ modus tollens we get

• $L' \quad \to (6)$

From $1$ and $6$ modus tollens we get

• $F'$

Ans is (b).  Bank receipt is not forged.

Comments

where are variable ??

Answer 2

Also solved using resolution principle

Answer 3

• $P:$ Bank receipt is forged
• $Q:$ Mr. $M$ is liable
• $R:$ He will go bankrupt
• $S:$ Bank loan him money

 

1. If the bank receipt is forged, then Mr. M is liable.   $P\rightarrow Q$
2. If Mr. M is liable, he will go bankrupt.    $Q\rightarrow R$

 

Now, With Hypothetical Syllogism 

$P\rightarrow Q$

$Q\rightarrow R$

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$P\rightarrow R$         -----------------$(1)$

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     $3.$If the bank will loan him money, he will not go bankrupt.    $S\rightarrow \sim R=\sim S\vee \sim R$-----------$(2)$

 

Now with Resolution  

and from $(1)$ and $(2)$

$\sim P\vee R$

$\sim S\vee \sim R$

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$\sim P\vee \sim S$---------------$(3)$

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      $4.$The bank will loan him money.    $S$

 Now with Disjunctive Sillogism  

$\sim P\vee \sim S$

$S$

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$\sim P$

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So, Conclusion is The bank receipt is not forged

Answer 4

Let us denote sentences with variables
F : Bank receipt is forged
L : Mr M is liable
B : He will go bankrupt
M : Bank loan him money
 
"If the bank receipt is forged, then Mr. M is liable" :  F->L ..(1)
"If Mr. M is liable, he will go bankrupt" : L->B ...(2)
"If the bank will loan him money, he will not go bankrupt" : M->B' ...(3)
"The bank will loan him money" : M ...(4)

optuions are :
a. "Mr. M is liable" : L
b. "The receipt is not forged" : F'
c. "Mr. M will go bankrupt" : B
d. "The bank will go bankrupt" :  NOT GETTING THIS
e. None of the above

From (3) and (4) , using  modus ponens we get
B' ...(5)
From (2) and (5) using modus tollens we get
L' ...(6)
From (1) and (6) using modus tollens we get
F' which is equivalent to option B.

Hence correct ans is B