Mathematics

Set Theory and Relations

368 Questions

Set theory involves the study of collections of objects and includes operations like union, intersection, and finding complements. Questions cover power sets, Cartesian products, and properties of empty sets. This foundational mathematical topic frequently appears in various competitive exams and university entrance tests.

set operationspower setscartesian productsset complementsproperties of empty setsset partitions

Set Theory and Relations Questions

Multiple choice

Which of the following is TRUE?

  1. Every subset of a regular set is regular.

  2. Every finite subset of a non-regular set is regular.

  3. The union of two non-regular sets is not regular.

  4. Infinite union of finite sets is regular.
Reveal answer Fill a bubble to check yourself
B Correct answer
Explanation

Multiple choice

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

x$\left [ i \right ]=\begin {cases} 1 & \text{if } i \in X \\ 0 & otherwise \end{cases}$ 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 = 0 ; i< n; ++i) z[i] = (x[i] Ù ~y[i]) Ú (~x[i] Ù y[i]) }

The set Z computed by the algorithm is

  1. (X U Y)
  2. (X $\cap$Y)
  3. (X − Y) $\cap$ (Y − X)

  4. (X − Y) U (Y − X)
Reveal answer Fill a bubble to check yourself
D Correct answer
Explanation

$\text{Here}\ z = (x \land y') \lor (x' \land y)$

Multiple choice

The subset-sum problem is defined as follows. Given a set of n positive integers, S = {a1, a2, a3, …, an}, and positive integer W, is there a subset of S whose elements sum to W? A dynamic program for solving this problem uses a 2-dimensional Boolean array, X, with n rows and W+1 columns. X [i, j], 1$\le$i $\le$ n,0 $\le$j $\le$ W, is TRUE if and only if there is a subset of {a1, a2, …, a} whose elements sum to j.

Which entry of the array X, if TRUE, implies that there is a subset whose elements sum to W?

  1. X[1, W]

  2. X[n, 0]

  3. X[n, W]

  4. X[n - 1, n]
Reveal answer Fill a bubble to check yourself
C Correct answer
Explanation

Multiple choice

Let A be a set having 'n' elements. The number of binary operations that can be defined on A, is
  1. $n^{n^2}$
  2. $2^{n^n}$
  3. $n^{2^n}$
  4. $2^{2^n}$
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

By definition, a binary operation defined on a set Y is a function $F : Y X Y \rightarrow ……..$ The domain, i.e. Y X Y has n × n elements (because Y has n elements). Each of ……… elements can be mapped to one of the 'n' elements of Y. So, totally $n^{n^2}$ binary is possible.

Multiple choice

Let x denote a real number. Find out the INCORRECT statement from the following, where a and b are real numbers.

  1. S = {x : x > 3} represents the set of all real numbers greater than 3.

  2. S = {x : x2 < 0} represents the empty set.
  3. S = {x : x $\in$A and x $\in$ B} represents the union of set A and set B.

  4. S = {x : a < x < b} represents the set of all real numbers between a and b.
Reveal answer Fill a bubble to check yourself
C Correct answer
Explanation

$\text{The incorrect statement is, S={$x : x\in \text{A and $x$}\in B$} represents the union of set A and set B.}\\ \text{The above symbol $(\in)$ denotes intersection of set A and set B. Therefore this statement is incorrect.}$

Multiple choice

If A = {Multiple of 3 between 1 and 20} and B = {Odd natural number less than 14}, which of the following is the union of two sets A and B?

  1. {1, 3, 5, 6, 6, 9, 11, 12, 13, 15, 18}

  2. {1, 3, 5, 6, 7, 9, 11, 12, 13, 15, 9}

  3. {1, 3, 5, 6, 7, 9, 11, 12, 13, 15, 18}

  4. {1, 3, 5, 6, 7, 9, 1, 12, 13, 15, 18}

  5. {1, 3, 5, 6, 9, 9, 11, 12, 13, 15, 18}
Reveal answer Fill a bubble to check yourself
C Correct answer
Explanation

As per the given information: A = {3, 6, 9, 12, 15, 18} and B = {1, 3, 5, 7, 9, 11, 13} Hence, the union of sets A and B is {1, 3, 5, 6, 7, 9, 11, 12, 13, 15, 18}.

Multiple choice mathematics and statistics set language de morgan's law for set theory complement of sets different sets de morgan's law

If $A$ and $B$ are subsets of $U$ such that $n(U) = 700, n(A) = 200, n(B) = 300, n$$\displaystyle \left ( A\cap B \right )$ $= 100$, then find $n\displaystyle \left ( A'\cap B' \right )$

  1. $405$
  2. $305$
  3. $400$
  4. $300$
Reveal answer Fill a bubble to check yourself
D Correct answer
Explanation

We know that $\displaystyle \left ( A\cup B \right )'=$ $\displaystyle A'\cap B'$ so we need to find $\displaystyle \left ( A\cup B \right ) $  first.
$n\displaystyle \left ( A\cup B \right ) =$ $n(A) + n(B) - n$$\displaystyle (A\cap B)$ $= 200 + 300 - 100 = 400$
$n\displaystyle \left ( A\cup B \right )' =$ $n(U) - n$$\displaystyle \left ( A\cup B \right )'=$$  700 - 400 = 300$
$\displaystyle \Rightarrow $ $n\displaystyle (A'\cap B')$$= 300$.