Set Theory in Discrete Mathematics Quiz

A binary operation ⊕ on a set of integers is defined as x ⊕ y = x² + y². Which one of the following statements is TRUE about ⊕?

What is the possible number of reflexive relations on a set of 5 elements?
 

We are given a set X = {x1, .... xn} where xi = 2ⁱ. A sample S ⊆ X is drawn by selecting each xi independently with probability pi = 1/2. The expected value of the smallest number in sample S is:

Let R be the set of all binary relations on the set {1, 2, 3}. Suppose a relation is chosen from R at random. The probability that the chosen relation is reflexive (round off to 3 decimal places) is ________ .

Note - This question was Numerical Type.

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 pr be the probability that an element (x,y,z) ∈ L³ chosen equiprobably satisfies x ∨ (y ∧ z) = (x ∨ y) ∧ (x ∨ z). Then

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

Note - This question was Numerical Type.

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


Note - This was Numerical Type question.

Let X and Y denote the sets containing 2 and 20 distinct objects respectively and F denote the set of all possible functions defined from X and Y. Let f be randomly chosen from F. The probability of f being one-to-one is _________

The number of onto functions (surjective functions) from set X = {1, 2, 3, 4} to set Y = {a, b, c} is ________________

Let N be the set of natural numbers. Consider the following sets, P: Set of Rational numbers (positive and negative) Q: Set of functions from {0, 1} to N R: Set of functions from N to {0, 1} S: Set of finite subsets of N Which of the above sets are countable?