Mathematics for Data Science-1 Week 1 Graded Assignment

Mathematics For Data Science-1 Week 1 Graded Assignment

1 Multiple Select Questions (MSQ)

Question 1 Question Which of the following√ is (are)√ rational number(s)? (3 + 3 5)(2 − 2 5) √ √64 25 √ 3+√5 3− 5

1 2 3

Question 2 Question Upendra has two daughters (Sukhalata and Punyalata) and one son (Sukumar). Sukumar has one son named Satyajit. Punyalata has two daughters (Kalyani and Nalini). This family tree has been shown in the ﬁgure below. Let us deﬁne a relation R as follows,

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

• S := {(A, B)|A is son of B}.

1 Upendra

Sukhalata Sukumar Punyalata

Satyajit Kalyani Nalini

Which of the following is (are) true? (Satyajit, Sukumar) ∈ S but (Sukumar, Satyajit) ∈/ S.

R is transitive relation.

(Sukumar, Upendra) ∈ S and (Satyajit , Sukumar) ∈ S but (Satyajit, Upendra) ∈/ S.

(Satyajit, Sukhalata) is neither in R nor in S.

Question 3 Question If A and B are sets and A ∪ B = A ∩ B then which of the following is(are) true? Either one of A or B is empty set.

B is proper subset of A.

A = B.

A is proper subset of B.

Question 4 Question Let R be a relation on a collection of sets deﬁned as follows,

R = {(A, B) | A ⊆ B}

Then pick out the correct statement(s). R is reﬂexive and transitive.

R is symmetric.

R is antisymmetric.

R is reﬂexive but not transitive.

Page 2 Question 5 Question Let a relation R be deﬁned as R = {(A, B) | Both A and B live in the same city}. Pick out the correct statement(s). R is symmetric. R is anti-symmetric. R is transitive. R is reﬂexive.

Question 6 Question Let X be the set of natural numbers divisible by 100, Y be the set of natural numbers divisible by 25, Z be the set of natural numbers which are perfect squares. Now consider the following Venn diagram. [Note: A, B, C, D, E, F, G are the region marked in the following Venn diagram (A region in the Venn-diagram can be an empty). For example, A = X\(Y ∪Z), B = Y \(X∪Z), and C = Z \ (X ∪ Y ).]

X Y B A D G F E C Z

Fig A-1.1

Choose the correct option(s) from below. 625 is in E. G is an empty set. 200 is in G. 2500 is in F . 2500 is in G. 25 is in F . A and F are empty.

Page 3 Question 7 Question Let L denote the following list of words: University, Institute, Company, Science, Literature, Movies, Computer, Technology. Let R and S be relations deﬁned on the set of pair of words formed from L as given below. • A pair of words are in R if they have atleast two distinct letters in common. • A pair of words are in S if they have atleast three distinct letters in common. For example, (Movies, Science) is both in R and S, as the letters s,i and e are in common. (University, Institute) is in both R and S. (Science, Literature) is in R but not in S. (Company, Computer), (Movies, Technology), (Institute, Literature) are all in S. There is a pair which is neither in R nor in S.

2 Multiple Choice Questions (MCQ):

Question 1 Question Consider the following table

Weight Name of students (in kg.) Akash 62 Hasan 55 Rahul 55 Jyoti 51 Saﬁna 54

Table A-1.1

We can think of this as a function f from the set of students to the set of integers between 50 and 70. Now pick out the correct statement from the following. f is one to one but not onto. f is onto but not one to one. f is neither one to one nor onto. f is bijective.

Page 4 Question 2 Question Suppose A = {a, b, c, d}. How many subsets of 2 distinct elements are possible? 2

Question 3 Question Let us consider the following sets,

• A = {x ∈ N | x mod 3 = 0 and 1 ≤ x ≤ 13}

• B = {x ∈ N | x mod 4 = 0 and 1 ≤ x ≤ 13}

• C = {x ∈ N | x mod 5 = 0 and 1 ≤ x ≤ 13} Which of the following Venn diagrams is accurate for these sets.

Diagram A Diagram B A B A B

C C

Fig A-1.2 Fig A-1.3

Diagram C Diagram D C B A B

A C

Fig A-1.4 Fig A-1.5

Page 5 Question 4 Question If A and B are sets then ﬁnd A ∩ (B ∪ A)c from the given Venn diagram

A B

Fig A-1.6

Ac ∪ Bc

Empty set

A ∪ B

Question 5 Question In a class of 120 students numbered 1 to 120, all even numbered students opt for P hysics, those whose numbers are divisible by 5 opt for Chemistry and those whose numbers are divisible by 7 opt for Mathematics. How many opt for none of the three subjects? 19

Question 6 Question Let us deﬁne a function f : Z → Z as follows, ( x , if x is even f(x) = 2 0, if x is odd

onto but not one to one.

one to one but not onto.

Page 6 one to one and onto.

neither one to one nor onto.

Page 7