A binary operation on a set of integers is defined as x y = x^{2} + y^{2}. Which one of the following statements is TRUE about ?
Consider the set S = {1, ω, ω2}, where ω and w^{2} are cube roots of unity. If * denotes the multiplication operation, the structure (S, *) forms
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Which one of the following in NOT necessarily a property of a Group?
Consider the binary relation R = {(x, y), (x, z), (z, x), (z, y)} on the set {x, y, z}. Which one of the following is TRUE?
For the composition table of a cyclic group shown below
Q.
Which one of the following choices is correct?
Let S be a set of nelements. The number of ordered pairs in the largest and the smallest equivalence relations on S are:
How many different nonisomorphic Abelian groups of order 4 are there
Consider the set S = {a,b,c,d}. consider the following 4 partitions π_{1}, π_{2}, π_{3}, π_{4} on S : π_{1} = π_{2} = π_{3} = π_{4} = Let ρ be the partial order on the set of partitions S' = {π_{1},π_{2},π_{3}, π_{4}} defined as follows : π_{i } ρ π_{j if and only if }π_{i }refines πj . The poset diagram for (S', ρ) is :
Consider the set of (column) vector defined by
Which of the following is True ?
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:
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?
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: Then R is:
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}.
Q.
The value of Log_{2}Log_{2}N is ______.
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.
Q. Which one of the following is CORRECT?
Let X and Y be finite sets and f: X > Y be a function. Which one of the following statements is TRUE?
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.
Q. The size of L is __________.
If V1 and V2 are 4dimensional subspaces of a 6dimensional vector space V, then the smallest possible dimension of V1 ∩ V2 is ______.
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
Q.
where e is the identity element. The maximum number of elements in such a group is __________.
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.
Q.
Which one of the following is CORRECT?
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?
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?
Let S = {1,2,3, ....... , m}, m >3. Let X_{1} ...... X_{n} be subset of S each of size 3. Define a function f from S to the set of atural numbers as, f(i) is the number of sets X_{j } that contain the element i. That is
Let A, B and C be nonempty sets and let X = (A  B)  C and Y = (A  C)  (B  C). Which one of the following is TRUE?
The following is the Hasse diagram of the poset [{a, b, c, d, e}, ≤]
The poset is
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
Let R and S be any two equivalence relations on a nonempty set A. Which one of the following statements is TRUE?
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?
What is the minimum number of ordered pairs of nonnegative 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"
Consider the binary relation:
S = {(x, y)  y = x+1 and x, y ∈ {0, 1, 2, ...}}
Q. The reflexive transitive closure of S is
55 docs215 tests

55 docs215 tests
