Let, 

Let A={1,2,3,4,5,6,7} and B = {1,2,3, $$\cdots$$​ 10} Then, the number of injective functions

from A to B is?

An infinite binary word a is a string (a₁a₂a₃ ...), where each aₙ ∈ {0,1}.

Fix a word s = (s₁s₂s₃ ...), where sₙ = 1 if and only if n is prime.

Let S = {a = (a₁a₂a₃ ...) | ∃m ∈ ℕ such that aₙ = sₙ, ∀n ≥ m}.

What is the cardinality of S?

Let $X$ be a non-empty set and $P(X)$ be the set of all subsets of $X$. On $P(X)$, define two operations $\ast$ and $\Delta$ as follows: for $A,B\in P(X)$,

• $A\ast B=A\cap B$ (Intersection of sets)
• $A\Delta B=(A\cup B)\setminus (A\cap B)$ (Symmetric difference of sets)

Which of the following statements is true?

Suppose $$f:[0,1]\times[0,1] \to (0,1)\times (0,1)$$ is a continuous,

non constant function. Which of the following statements 

are NOT true ?

Let X, Y be defined by

​$$X = \{(x_n)_{n \geq 1} : \limsup_{n \to \infty} x_n = 1, \; \text{where} \; x_n \in \{0, 1\}\}$$​​

and

​$$Y = \{(x_n)_{n \geq 1} : \lim_{n \to \infty} x_n \; \text{does not exist, where} \; x_n \in \{0, 1\}\}$$​.

Which of the following is true?

Suppose that A and B are two non-empty subsets of  $$\R$$ and, $$C=A \cap B$$ .

Which of the following conditions imply that C is empty . 

Which of the following statements is true ?

​$$\text{Let } \{E_n\} \text{ be a sequence of subsets of } \mathbb{R}. \\\text{Define }\\\limsup_{n} E_n = \bigcap_{k=1}^\infty \bigcup_{n=k}^\infty E_n\\\liminf_{n} E_n = \bigcup_{k=1}^\infty \bigcap_{n=k}^\infty E_n\\\text{Which of the following statements is true?}$$​​

​​​

​Which of the following sets are countable ?​