Question 1

**Section I (14 Marks)** 1. In the context of real numbers and the standard order relation, which of the following statement(s) is (are) true? [2 marks] (a) The least upper bound of a set is always an element of the set.\ (b) A set may have more than one least upper bound.\ (c) Every finite set has a least upper bound.\ (d) Every subset of $[0, 1]$ has a least upper bound. 2. Which of the following statements is/are equivalent to the negation of the following statement? [4 marks] $$(p|a \text{ and } p|b) \text{ or } p|c$$ (a) $(p \nmid a \text{ or } p \nmid b) \text{ and } p \nmid c$.\ (b) $(p \nmid a \text{ and } p \nmid c) \text{ or } (p \nmid b \text{ and } p \nmid c)$.\ (c) $(p \nmid a \text{ and } p \nmid b) \text{ or } p \nmid c$.\ (d) $(p \nmid a \text{ or } p \nmid c) \text{ and } (p \nmid b \text{ or } p \nmid c)$. 3. Which of the following options is/are true? [4 marks] (a) For every convergent sequence $\{s_n\}$ in $\mathbb{R}$ such that $s_n > 0$ for all $n$, $\lim\limits_{n \to \infty} s_n > 0$\ (b) For any sequence $\{s_n\}$ in $\mathbb{R}$, if $\lim\limits_{n \to \infty} |s_n| = 0$, then $\lim\limits_{n \to \infty} s_n = 0$\ (c) For any sequence $\{s_n\}$ in $\mathbb{R}$, if $\lim\limits_{n \to \infty} s_n = 0$, then $\lim\limits_{n \to \infty} |s_n| = 0$\ (d) For any sequence $\{s_n\}$ in $\mathbb{R}$, if $\{|s_n|\}$ is convergent then $\{s_n\}$ is convergent. 4. Which of the following options is/are true? [4 marks] (a) Set $\{n \in \mathbb{N} \mid n \geq 5\}$ and set $\{n \in \mathbb{N} \mid n \geq 500\}$ have the same cardinality.\ (b) $\displaystyle\sum_{k=1}^{n}(3k - 1) = \frac{n(3n+1)}{2}$ for every natural number $n$.\ (c) If $S$ is the set $\{1, 2, 3, 4\}$, then cardinality of the set $(S \times S) - (S \times \{3\})$ is 12.\ (d) For every positive integer $n$, $\sum_{i=1}^{n}\sum_{j=1}^{i} \frac{i}{i+j} = \sum_{j=1}^{n}\sum_{i=j}^{n} \frac{i}{i+j}$. **Section II (26 Marks)** 1. Consider three sets $A, B$ and $C$. Prove that $A \times (B \cup C) = (A \times B) \cup (A \times C)$. [6 marks] 2. The Fibonacci sequence $1, 1, 2, 3, 5, 8, 13, 21, 34, \ldots$ is defined recursively by $a_1 = 1$, $a_2 =1$, and $a_{n+2} = a_{n+1} + a_n$, for all $n \geq 1$. Prove that $\gcd(a_k, a_{k+1}) = 1$ for all $k \in \mathbb{N}$. [6 marks] 3. Prove or disprove the following statement: For any non-empty subset $A$ of $\mathbb{Q}$ that is bounded from above, there exists a least upper bound for $A$ in the set of rational numbers $\mathbb{Q}$. [6 marks] 4. Consider a sequence $\{a_n\}$ such that $a_1 = 4$ and $a_{n+1} = 4^{a_n}$, for all $n > 1$. Find the remainder when $a_{100}$ is divided by 7. [8 marks]