568 views
• 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?

1. Mr. M is liable
2. The receipt is not forged
3. Mr. M will go bankrupt
4. The bank will go bankrupt
5. None of the above

edited | 568 views

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.

by Boss (31.4k points)
edited by
0

where are variable ??

• $P:$ 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

$P\rightarrow Q$

$Q\rightarrow R$

----------------------

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

-----------------------

$3.$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 $(2)$

$\sim P\vee R$

$\sim S\vee \sim R$

------------------------

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

-------------------------

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

Now with Disjunctive Sillogism

$\sim P\vee \sim S$

$S$

--------------------------

$\sim P$

-------------------------

So, Conclusion is The bank receipt is not forged

by Veteran (119k points)
edited by
+1 vote
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
by Active (2.8k points)
+1 vote

Also solved using resolution principle by Loyal (5.3k points)