Discrete Mathematics
Q281.
Consider the field C of complex numbers with addition and multiplication. Which of the following form(s) a subfield of C with addition and multiplication? S1: the set of real numbers S2:\{(a + ib) \mid a and b are rational numbers\} S3:\{a + ib \mid (a^2 + b^2) \leq 1\} S4: \{ia \mid a \text{ is real}\}Q282.
For the set N of natural numbers and a binary operation f : N \times N \to N, an element z \in N is called an identity for f, if f (a, z) = a = f(z, a), for all a \in N. Which of the following binary operations have an identity? i. f (x, y) = x + y - 3 ii. f (x, y) = \max(x, y) iii. f (x, y) = x^yQ283.
A set X can be represented by an array x[n] as follows x[i]=\left\{\begin{matrix} 1 &if i\in X \\ 0& otherwise \end{matrix}\right. Consider the following algorithm in which x,y and z are boolean arrays of size n; algorithm zzz(x[] , y[], z []) { int i; for (i=O; i < n; ++i) z[i] = (x[i] ^ ~y[i]) V (~x[i] ^ y[i]) } The set Z computed by the algorithm isQ284.
Given a set of elements N = {1,2,...,n} and two arbitrary subsets A\subseteqN and B\subseteqN , how many of the n! permutations p from N to N satisfy min[p(A)]=min[p(B)], where min(S) is the smallest integer in the set of integers S and p(S) is the set of integers obtained by applying permutation p to each element of S ?Q286.
We are given a set X=\{x_{1},...,x_{n}\} where x_{i}=2^{i}. A sample S\subseteq X is drawn by selecting each x_{i} independently with probability p_{i}=\frac{1}{2}. The expected value of the smallest number in sample S is:Q287.
Let A be a set with n elements. Let C be a collection of distinct subsets of A such that for any two subsets S_1 and S_2 in C, either S_1 \subset S_2 or S_2\subset S_1. What is the maximum cardinality of C?Q288.
Let S={1,2,3....,m},m \gt 3. Let X_{1},...,X_{n} be 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 X_{j} that contain the element i. That is f(i)=|\{j|i\in X_{j}\}|. Then \sum_{i=1}^{m}f(i)Q289.
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?Q290.
Let n = p^{2}q, where p and q are distinct prime numbers. How many numbers m satisfy 1 \leq m \leq n and gcd(m,n)=1? Note that gcd(m,n) is the greatest common divisor of m and n.