Set Theory Practice Questions

If P, Q, R are subsets of the universal set U, then $(P\cap Q\cap R)\cup (P^{C} \cap Q \cap R)\cup Q^{C} \cup R^{C}$ is

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: $\{ia \mid a \text{ is real}\}$

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?

Which of the following is TRUE?

The set of all Equivalence Classes of a set A of Cardinality C

Let S be a set of n elements. The number of ordered pairs in the largest and the smallest equivalence relations on S are:

A set X can be represented by an array x[n] as follows

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 is

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 $\sum_{i=1}^{m}f(i)$

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:

Given a set of elements N = {1,2,...,n} and two arbitrary subsets A $\subseteq$ N and B $\subseteq$ N , 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 ?

Let E,F and G be finite sets. Let X=(E $\cap$ F) - (F $\cap$ G) and Y = (E - (E $\cap$ G)) - (E - F). Which one of the following is true?

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. $f (x, y) = x^y$

Let X, Y, Z be sets of sizes x, y and z respectively. Let W=X $\times$ Y and E be the set of all subsets of W. The number of functions from Z to E is

What is the cardinality of the set of integers X defined below? $X=\{n \mid 1 \leq n \leq 123$ , n is not divisible by either 2, 3 or 5}

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.

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?

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?

In a class of 200 students, 125 students have taken Programming Language course, 85 students have taken Data Structures course, 65 students have taken Computer Organization course; 50 students have taken both Programming Language and Data Structures, 35 students have taken both Programming Language and Computer Organization; 30 students have taken both Data Structures and Computer Organization, 15 students have taken all the three courses.How many students have not taken any of the three courses?