8 votes

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?

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

16 votes

Best answer

Let's name the events:

**$\mathbf{S}$** = Abhay gets **s**hoes

**$\mathbf{N}$** = Asha gets a **n**ecklace

**$\mathbf{T}$** = Arun gets a **T**-shirt

**$\mathbf{B}$** = Aditi gets **b**angles

**$\mathbf{M}$** = **M**other 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)**