Set Theory & Algebra

Question 21

Let E, F and G be finite sets. Let X = (E ∩ F) - (F ∩ G) and Y = (E - (E ∩ G)) - (E - F). Which one of the following is true?

• X ⊂ Y

• X ⊃ Y

• X = Y

• X - Y ≠ φ and Y - X ≠ φ

Question 22

Given a set of elements N = {1, 2, ..., n} and two arbitrary subsets A⊆N and B⊆N, how many of the n! permutations π from N to N satisfy min(π(A)) = min(π(B)), where min(S) is the smallest integer in the set of integers S, and π(S) is the set of integers obtained by applying permutation π to each element of S?

• (n - |A ∪ B|) |A| |B|
• (|A|²+|B|²)n²
• n! |A∩B| / |A∪B|
• |A∩B|²nC|A∪B|

Question 23

Let S = {1, 2, 3, ...., m}, m>3. Let x1, x2,....xn be the subsets of S each of size 3. Define a function f from S to the set of natural numbers as, f(i) is the number of sets [Tex]X_j[/Tex] that contain the element i. That is, f(i) = |{j|i [Tex]\\epsilon[/Tex] [Tex]X_j[/Tex]}|. Then, [Tex]\\sum_{i=1}f(i)[/Tex] is :

• 3m
• 3n
• 2m + 1
• 2n + 1

Question 24

Let A, B and C be non-empty sets and let X = (A - B) - C and Y = (A - C) - (B - C). Which one of the following is TRUE?

• X = Y

• X ⊂ Y

• Y ⊂ X

• none of these

Question 25

The following is the Hasse diagram of the poset [{a, b, c, d, e}, ≤] The poset is

• not a lattice
• a lattice but not a distributive lattice
• a distributive lattice but not a Boolean algebra
• a Boolean algebra

Question 26

The set {1, 2, 4, 7, 8, 11, 13, 14} is a group under multiplication modulo 15. The inverses of 4 and 7 are respectively

• 3 and 13
• 2 and 11
• 4 and 13
• 8 and 14

Question 27

Let R and S be any two equivalence relations on a non-empty set A. Which one of the following statements is TRUE?

• R ∪ S, R ∩ S are both equivalence relations

• R ∪ S is an equivalence relation

• R ∩ S is an equivalence relation

• Neither R ∪ S nor R ∩ S is an equivalence relation

Question 28

Let f: B → C and g: A → B be two functions and let h = f o g. Given that h is an onto function. Which one of the following is TRUE?

• f and g should both be onto functions.
• f should be onto but g need not be onto
• g should be onto but f need not be onto
• both f and g need not be onto

Question 29

What is the minimum number of ordered pairs of non-negative numbers that should be chosen to ensure that there are two pairs (a, b) and (c, d) in the chosen set such that "a ≡ c mod 3" and "b ≡ d mod 5"

• 4
• 6
• 16
• 24

Question 30

Consider the binary relation:

S = {(x, y) | y = x+1 and x, y ∈ {0, 1, 2, ...}}

The reflexive transitive closure of S is

• {(x, y) | y > x and x, y ∈ {0, 1, 2, ... }}
• {(x, y) | y ≥ x and x, y ∈ {0, 1, 2, ... }}
• {(x, y) | y < x and x, y ∈ {0, 1, 2, ... }}
• {(x, y) | y ≤ x and x, y ∈ {0, 1, 2, ... }}