Mathematical Thinking Quiz 1: 26 October 2025 (September 2025 term)

Section I (16 marks)

MSQ (4 marks)

1. Which of the following options is/are true for every function $f : \mathbb{N} \cup \{0\} \to \mathbb{N} \cup \{0\}$ such that $f(m+1) = f(m) + f(1)$ for all $m \in \mathbb{N} \cup \{0\}$.

a. $f(0) = 0$.
b. $f(n) = nf(1)$ for all $n \in \mathbb{N}$.
c. $f(mn) = f(m)f(n)$ for all $m, n \in \mathbb{N}$.
d. If $f(1) = 0$ then $f(n) = 0$ for all $n \in \mathbb{N}$.

MSQ (4 marks)

1. Which of the following options is/are true?

a. If $x \in \mathbb{R}$ and $x > 0$, then there exists $n \in \mathbb{N}$ such that $n > x$.
b. If $x \in \mathbb{R}$ and $x > 0$, then there exists $n \in \mathbb{N}$ such that $0 < \dfrac{1}{n} < x$.
c. If $x \in \mathbb{R}$ and $x > 0$, then there exists $n \in \mathbb{N}$ such that $n - 1 \leq x < n$.
d. If $x \in \mathbb{R}$ and $x > 0$, then there exists $n \in \mathbb{N}$ such that $n^2 - (n-1)^2 > x$.

MSQ (4 marks)

1. Which of the following sets have the same cardinality as $[0, 1]$?

a. $(0, 1)$
b. $(-1, 1)$
c. $[-1, 1]$
d. The set $\mathbb{R}$ of all real numbers.

SA (4 marks)

1. Find the remainder when $2^{81} + 3^{81}$ is divided by 7.

Section II (24 marks)

Subjective (7 marks)

1. Let $A$ be a set of real numbers bounded above and let $a = \sup A$. Let $B$ be the set whose elements are double those of $A$, i.e.,

$B = \{2x \mid x \in A\}$

Prove that $\sup B = 2a$.

Subjective (7 marks)

1. The Fibonacci sequence, denoted by $F_n$, is defined by the following recurrence relation and initial conditions:

$\begin{aligned} F_1 &= 1 \\ F_2 &= 1 \\ F_n &= F_{n-1} + F_{n-2} \quad \text{for } n \geq 3 \end{aligned}$

Thus the first few terms of the sequence are $1, 1, 2, 3, 5, 8, \ldots$

Prove the following identity using Mathematical Induction for all integers $n \geq 1$:

$\sum_{k=1}^{n} F_k = F_{n+2} - 1$

Subjective (10 marks)

1. Prove the statement: Let $a, b, c \in \mathbb{Z}$. The linear equation $ax + by = c$ has an integer solution $(x, y) \in \mathbb{Z} \times \mathbb{Z}$ if and only if $c$ is divisible by $\gcd(a, b)$.