Opening the paper…
Mathematical Thinking, Quiz 1
Section I (16 marks)
MSQ (4 marks)
a. .
b. for all .
c. for all .
d. If then for all .
MSQ (4 marks)
a. If and , then there exists such that .
b. If and , then there exists such that .
c. If and , then there exists such that .
d. If and , then there exists such that .
MSQ (4 marks)
a.
b.
c.
d. The set of all real numbers.
SA (4 marks)
**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) 2. 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) 3. 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) 4. Find the remainder when $2^{81} + 3^{81}$ is divided by 7. **Section II (24 marks)** Subjective (7 marks) 5. 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) 6. 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) 7. **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 $\gcd(a, b)$.