Set Theory & Algebra

Question 11

Let X, Y, Z be sets of sizes x, y and z respectively. Let W = X x Y. Let E be the set of all subsets of W. The number of functions from Z to E is:

• z

  ²

  ^xy

• z x 2

  ^xy

• z

  ²

  ^{x + y}

• 2

  ^xyz

Question 12

The set {1, 2, 3, 5, 7, 8, 9} under multiplication modulo 10 is not a group. Given below are four plausible reasons. Which one of them is false?

• It is not closed
• 2 does not have an inverse
• 3 does not have an inverse
• 8 does not have an inverse

Question 13

A relation R is defined on ordered pairs of integers as follows: (x,y) R(u,v) if x < u and y > v. Then R is:

• Neither a Partial Order nor an Equivalence Relation

• A Partial Order but not a Total Order

• A Total Order

• An Equivalence Relation

Question 14

Let S denote the set of all functions f: {0,1}⁴ -> {0,1}. Denote by N the number of functions from S to the set {0,1}. The value of Log₂Log₂N is ______.

• 12
• 13
• 15
• 16

Question 15

Consider the following relation on subsets of the set S of integers between 1 and 2014. For two distinct subsets U and V of S we say U < V if the minimum element in the symmetric difference of the two sets is in U. Consider the following two statements:

S1: There is a subset of S that is larger than every other subset.
S2: There is a subset of S that is smaller than every other subset. 

Which one of the following is CORRECT?

• Both S1 and S2 are true

• S1 is true and S2 is false

• S2 is true and S1 is false

• Neither S1 nor S2 is true

Question 16

Let X and Y be finite sets and f: X -> Y be a function. Which one of the following statements is TRUE?

• A

• B

• C

• D

Question 17

Let G be a group with 15 elements. Let L be a subgroup of G. It is known that L != G and that the size of L is at least 4. The size of L is __________.

• 3
• 5
• 7
• 9

Question 18

If V1 and V2 are 4-dimensional subspaces of a 6-dimensional vector space V, then the smallest possible dimension of V1 ∩ V2 is ______.

• 1
• 2
• 3
• 4

Question 19

There are two elements x, y in a group (G,∗) such that every element in the group can be written as a product of some number of x\'s and y\'s in some order. It is known that

  x ∗ x = y ∗ y = x ∗ y ∗ x ∗ y = y ∗ x ∗ y ∗ x = e

where e is the identity element. The maximum number of elements in such a group is __________.

• 2
• 3
• 4
• 5

Question 20

Consider the set of all functions f: {0,1, … ,2014} → {0,1, … ,2014} such that f(f(i)) = i, for all 0 ≤ i ≤ 2014. Consider the following statements:

P. For each such function it must be the case that 
   for every i, f(i) = i.
Q. For each such function it must be the case that 
   for some i, f(i) = i.
R. Each such function must be onto. 

Which one of the following is CORRECT?

• P, Q and R are true
• Only Q and R are true
• Only P and Q are true
• Only R is true