Mathematical Thinking Quiz 1: 7 July 2024 (May 2024 term)

1 Section I (15 Marks)

1. Which of the following options is/are true? [Marks: 4]

(a) If $A$ and $B$ are subsets of $\mathbb{R}$ such that $A \subset B$ and $B$ is countable, then $A$ is also countable.

(b) The set of irrational numbers is countable.

(c) If a set $X$ satisfies Peano's axioms, then there is a bijection between the set of natural numbers and the set $X$.

(d) $\displaystyle\sum_{i=1}^{4}\sum_{j=1}^{3} ij = 59$

1. Which of the following options is/are true? [Marks: 4]

(a) Set $\{5^n \mid n \in \mathbb{N}\}$ has an upper bound.

(b) 437 is a prime number.

(c) The supremum of the set $\{1 - \frac{(-1)^n}{n} \mid n \in \mathbb{N}\}$ is 1.

(d) The negation of the statement “Both $X$ and $Y$ are false, or both $X$ and $Y$ are true” is“($X$ is false or $Y$ is false) and ($X$ is true or $Y$ is true)”.

1. In a survey of 500 people, it was found that 49% liked to watch comedy movies, 53% liked thriller movies, and 62% liked romantic movies. In addition, 27% liked to watch both comedy and thriller, 29% liked to watch both thriller and romantic, and 28% liked to watch both comedy and romantic. 5% did not like any of the above types. How many people liked all three types of movies? [Marks: 4]

2. Which of the following options is/are true? [Marks: 3]

(a) $1 + \sqrt{2} + \sqrt{5}$ is an irrational number.

(b) If $a$ and $b$ are two real numbers such that $a < b$, then there exist only finitely many real numbers $\alpha$ such that $a < \alpha < b$.

(c) Let $a, b \in \mathbb{N}$ such that $a^3 \mid b^3$, then $a \mid b$.

(d) Let $a, b \in \mathbb{N}$ be such that $a^a \mid b^b$, then $a \mid b$.

2 Section II (25 Marks)

1. Let $A$ and $B$ be bounded subsets of $\mathbb{R}^+$. Define a set $A+B = \{x+y \mid x \in A \text{ and } y \in B\}$. Prove the following:

(a) $A+B$ is bounded. [Marks: 3]

(b) $\sup(A+B) = \sup(A) + \sup(B)$ [Marks: 6]

1. Assume $\gcd(a, b) = 1$. Prove that $\gcd(a+b, a-b) = 1$ or 2. [Marks: 8]

2. Let $p$ and $q$ be distinct primes such that $a^p \equiv a(\text{mod } q)$ and $a^q \equiv a(\text{mod } p)$. Then prove that $a^{pq} \equiv a(\text{mod } pq)$. [Marks: 8]