Set Theory
Q22.
A partial order P is defined on the set of natural numbers as follows. Here \frac{x}{y} denotes integer division. i.(0, 0) \in P. ii.(a, b) \in P if and only if (a \% 10) \leq (b \% 10) and (\frac{a}{10},\frac{b}{10})\in P. Consider the following ordered pairs: i. (101, 22) ii. (22, 101) iii. (145, 265) iv. (0, 153) Which of these ordered pairs of natural numbers are contained in P?Q24.
If P, Q, R are subsets of the universal set U, then (P\cap Q\cap R)\cup (P^{C} \cap Q \cap R)\cup Q^{C} \cup R^{C} isQ25.
Let S be a set of n elements. The number of ordered pairs in the largest and the smallest equivalence relations on S are:Q26.
A binary operation \oplus on a set of integers is defined as x \oplus y= x^{2}+y^{2}. Which one of the following statements is TRUE about \oplus ?Q27.
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}\}Q28.
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^yQ29.
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 isQ30.
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 ?