CMI2017-A-03

Question

Four siblings go shopping with their father. If Abhay gets shoes, then Asha does not get a necklace. If Arun gets a T-shirt, then Aditi gets bangles. If Abhay does not get shoes or Aditi gets bangles, the mother will be happy. Which of the following is TRUE?

• A. If the mother is happy, then Aditi got bangles.
• B. If Aditi got bangles, then Abhay got shoes.
• C. If the mother is not happy, then Asha did not get a necklace and Arun did not get a T-shirt.
• D. None of the above.

Answer 1

Let's name the events:

$\mathbf{S}$ = Abhay gets shoes

$\mathbf{N}$ = Asha gets a necklace

$\mathbf{T}$ = Arun gets a T-shirt

$\mathbf{B}$ = Aditi gets bangles

$\mathbf{M}$ = Mother is happy.

Now understanding the questions in a logical way:

NOTE: "if p, then q" is equivalent to writing: $p \rightarrow \ q$

If Abhay gets shoes, then Asha does not get a necklace.

This can be written as:

$\mathbf{S \rightarrow \ \sim N }$

If Arun gets a T-shirt, then Aditi gets bangles. 

$\mathbf{T \rightarrow \ B }$

If Abhay does not get shoes or Aditi gets bangles, the mother will be happy. 

$\mathbf{(\sim S \vee B) \rightarrow \ M }$

We have to see that what can be inferred from these three statements.

Now looking at the options:

• A. If the mother is happy, then Aditi got bangles.
  $M \rightarrow \ B $
  This can not be inferred because it might be the case that Aditi does not get bangles, but the mother is still happy.
   
• B. If Aditi got bangles, then Abhay got shoes.
  $B \rightarrow \ S $
  This also can not be inferred. It can be that Aditi got bangles, but Abhay does not get shoes.
   
• C. If the mother is not happy, then Asha did not get a necklace and Arun did not get a T-shirt.
  $\sim M \rightarrow \ (\sim N \ \wedge \sim T) $
  This can be inferred as follows:
  We have: $(\sim S \vee B) \rightarrow \ M$
  Contrapositive of this is: $\sim M \rightarrow \ \sim (\sim S \vee B)$
  $\ \ =\ \sim M \rightarrow \ (S \wedge \sim B)$    ----- $(1)$
  We also have: $S \rightarrow \ \sim N$ ----- $(2)$,
  And: $T \rightarrow \ B$
  whose contrapositive is: $\sim B \rightarrow \ \sim T$ ----- $(3)$
  From $(1), (2)$ and $(3)$ we can conclude:
  $\sim M \rightarrow \ (\sim N \ \wedge \sim T) $

Hence, the correct option should be (C)

Comments

didnt get C part

Answer 2