TIFR-2018-Maths-B: 8

Question

Consider the following four sets of maps $f:\mathbb{Z}\rightarrow \mathbb{Q}$:

1. $\{f:\mathbb{Z}\rightarrow \mathbb{Q} \mid f$ is bijective and increasing$\}$,
2. $\{f:\mathbb{Z}\rightarrow \mathbb{Q} \mid f$ is onto and increasing$\}$,
3. $\{f:\mathbb{Z}\rightarrow \mathbb{Q} \mid f$ is bijective, and satisfies that $ \forall n\leq 0,f\left ( n \right )\geq 0\}$, and
4. $\{f:\mathbb{Z}\rightarrow \mathbb{Q} \mid f$ is onto and decreasing$\}$.

How many of these sets are empty?

• A. $0$
• B. $1$
• C. $2$
• D. $3$

Comments

$\forall$  Q $\exists$ Z is not true, so it can be empty set.