Question 1: Section I (16 marks) MSQ (4 marks) Which of the following…

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.