The number of subsets {1, 2, ... n} with odd cardinality is __________.

The number of elements in he power set P (S) of the set S = {(φ), 1, (2, 3)} is:

Let G be a group of 35 elements. Then the largest possible size of a subgroup of G other than G itself is ______.

Suppose A = {a,b,c,d} and Π₁ is the following partition of A
Π₁ = {{a,b,c}{d}}

(a) List the ordered pairs of the equivalence relations induced by Π₁.

(b) Draw the graph of the above equivalence relation.

(c) Let Π₂ = {{a},{b},{C},{d}}
Π₃ = {{a,b,c,d}}
and Π₄ = {{a,b},{c,d}}

Draw a Poset diagram of the poset,
({Π₁,Π₂,Π₃,Π₄}, refines⟩

Let S be an infinite set and S₁ ..., S_n be sets such that S₁ ∪ S₂ ∪ ... ∪ S_n = S. Then,

Let A be a finite set of size n. The number of elements in the power set of A × A is:

If the longest chain in a partial order is of length n then the partial order can be written as a ______ of n antichains.

Let n = p²q, where p and q are distinct prime numbers. How many numbers m satisfy 1 ≤ m ≤ n and gcd (m, n) = 1? Note that gcd (m, n) is the greatest common divisor of m and n.

Let A be a set with n elements. Let C be a collection of distinct subsets of A such that for any two subsets S₁ and S₂ in C, either S₁ ⊂ S₂ or S₂ ⊂ S₁. What is the maximum cardinality of C?

Let f be a function from a set A to a set B, g a function from B to C, and h a function from A to C, such that h(a) = g(f(a)) for all a ∈ A. Which of the following statements is always true for all such functions f and g?

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?

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

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 __________.

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

The cardinality of the power set of {0, 1, 2, … 10} is _________.

Let R be the relation on the set of positive integers such that aRb if and only if a and b are distinct and have a common divisor other than 1. Which one of the following statements about R is true?

Consider the operations f(X, Y, Z) = X'YZ + XY' + Y'Z'  and  g(X′, Y, Z) = X′YZ + X′YZ′ + XY Which one of the following is correct?

Suppose L = {p, q, r, s, t} is a lattice represented by the following Hasse diagram:

For any x, y ∈ L, not necessarily distinct, x ∨ y and x ∧ y are join and meet of x, y respectively. Let L³ = {(x,y,z): x, y, z ∈ L} be the set of all ordered triplets of the elements of L. Let p_r be the probability that an element (x,y,z) ∈ L³ chosen equiprobably satisfies x ∨ (y ∧ z) = (x ∨ y) ∧ (x ∨ z). Then

Consider the set X = {a,b,c,d e} under the partial ordering

R = {(a,a),(a,b),(a,c),(a,d),(a,e),(b,b),(b,c),(b,e),(c,c),(c,e),(d,d),(d,e),(e,e)}.

The Hasse diagram of the partial order (X,R) is shown below.

The minimum number of ordered pairs that need to be added to R to make (X,R) a lattice is _________.

Let G be a finite group on 84 elements. The size of a largest possible proper subgroup of G is _________.

Let R be a relation on the set of ordered pairs of positive integers such that ((p,q),(r,s)) ∈ R if and only if p - s = q - r. Which one of the following is true about R?