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:
- 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.
- 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.
- 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)