Week_1_Assignment

Week 1 - Graded Assignment 1 - Mathematics I

Question 1

Which of the following are irrational numbers?

• $(\sqrt{8}+\sqrt{2})(\sqrt{12}-\sqrt{3})$
• $(\sqrt{8}-\sqrt{2})(\sqrt{18}+\sqrt{2})$
• $\frac{\sqrt{6}+\sqrt{3}}{\sqrt{6}-\sqrt{3}}$
• $\frac{\sqrt{8}+\sqrt{2}}{\sqrt{8}-\sqrt{2}}$

Accepted Answers:

• $(\sqrt{8}+\sqrt{2})(\sqrt{12}-\sqrt{3})$
• $\frac{\sqrt{6}+\sqrt{3}}{\sqrt{6}-\sqrt{3}}$

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Question 2

Suppose $f:D⟶\mathbb{R}$ is a function defined by $f(x)=\frac{\sqrt{x^2-16}}{x+4}$, where $D\subset \mathbb{Z}$. Let $A$ be the set of integers which are not in the domain of $f$, then find the cardinality of the set $A$.

Accepted Answers:

• 8

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Question 3

Consider the set $S=\{a\mid a\in \mathbb{N},a\le 33\}$. Let $R_1\text{and}{R}_2$ be relations from $S$ to $S$ defined as $R_1=\{(x,y)\mid x,y\in S,y=3x\}$ and $R_2=\{(x,y)\mid x,y\in S,y=x^2\}$. Find the cardinality of the set $R_1\setminus (R_1\cap R_2)$.

Accepted Answers:

• 10

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Question 4

In a Zoo, there are 6 Bengal white tigers and $6$ Bengal royal tigers. Out of these tigers, 5 are males and 10 are either Bengal royal tigers or males. Find the number of female Bengal white tigers in the Zoo.

Accepted Answers:

• 2

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Question 5

A survey was conducted on pollution of 525 ponds across some cities. It was found that 230 ponds are polluted by fertilisers $(F)$, 245 ponds are polluted by pesticides $(P)$ and 257 ponds are polluted by pharmaceutical products $(Ph)$. 100 ponds are polluted by fertilisers and pesticides, 82 ponds are polluted by fertilisers and pharmaceutical products, 77 ponds are polluted by pesticides and pharmaceutical products.

Define a relation on the set of 525 ponds such that two ponds are related if both are polluted by fertilisers and pharmaceutical products. Which of the following is/are true?

• Relation is reflexive.
• Relation is transitive.
• Relation is symmetric.
• This is an equivalence relation.

Accepted Answers:

• Relation is transitive.
• Relation is symmetric.

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Question 6

Consider the following table of a few materials and their dielectric constant values.

Material	Dielectric constant
Air	1
Vaccum	2
Paper	3
Glass	8
Nerve membrane	7
Silicon	13

We can think of this as a function $f$ from the set of materials to the set of dielectric constant values consisting of the elements $\{1,2,3,8,7,13\}$. Now pick out the correct statement from the following.

• $f$ is neither one to one nor onto.
• $f$ is one to one but not onto.
• $f$ is onto but not one to one.
• $f$ is bijective.

Accepted Answers:

• $f$ is bijective.

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Question 7

Consider the following sets.

• $A=\{x\in \mathbb{N}\mid x\text{ mod }2=0\text{ and }1\le x\le 10\}$
• $B=\{x\in \mathbb{N}\mid x\text{ mod }5=0\text{ and }6\le x\le 25\}$
• $C=\{x\in \mathbb{N}\mid x\text{ mod }7=0\text{ and }7\le x\le 29\}$

What is the cardinality of $(A\setminus (B\cup C))\cup (B\setminus (C\cup A))\cup (C\setminus (B\cup A))$

Accepted Answers:

• 11

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Question 8

Mahesh has four sons (Shubh, Rabi, Mahendra, and Rajat). Shubh has two sons (Yashubh and Navrtna). Rabi has two sons named Rathi and Rakesh. This family tree is shown in the figure below. Let us define two relations, $R$ and $S$, on the set $M$, which is the collection of all family members, as follows,

$R:=\{(A,B)|A\text{ and }B\text{ are cousins, i.e. their parents are siblings}\}$.

$S:=\{(A,B)|A\text{ is son of }B\}$.

If $m$ is the cardinality of the set $R$ and $n$ is the cardinality of the set $S$, then find the value of $m+n$.

Accepted Answers:

• 16

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Question 9

Define a function $f:\mathbb{Q}\to \mathbb{Z}$, such that $f(p/q)=p-q$, where $gcd(p,q)=1$. Which of the following option(s) is(are) true?

• $f$ is one to one but not onto
• $f$ is neither one to one nor onto.
• $f$ is onto but not one to one.
• $f$ is a bijective function.

Accepted Answers:

• $f$ is onto but not one to one.

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Question 10

Suppose $f_1(x)$ and $f_2(x)$ are functions defined on domains $D_1\subset \mathbb{R}$ and $D_2\subset \mathbb{R}$, respectively and codomains are subset of $\mathbb{R}$ . What will be the domain of the function $f_1(x)+f_2(x)$?

• $D_1\cup D_2$
• $D_1\setminus D_2$
• $D_2\setminus D_1$
• $D_1\cap D_2$

Accepted Answers:

• $D_1\cap D_2$