Mathematics

Set Theory and Relations

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

For any two sets A and B, $\left { (A\setminus B)\cup (B\setminus A) \right }\cap (A\cap B)$ is:

  1. $\phi $
  2. $ A \cup B$
  3. $A \cap B$
  4. $A' \cap B'$
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

$(A$ \ $B)\cup(B$ \ $A)$ contains that elements form set $A$ and $B$ that are not contained in the other set.

$(A$ \ $B)\cup(B$ \ $A)=(A\cap B)$ \ $(A\cap B)$
$\therefore (A$ \ $B)\cup(B$ \ $A)$ does not contain elements of $A\cap B$
Hence {$(A$ \ $B)\cup(B$ \ $A)$}$\cap (A\cap B)=${$\phi$}

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

Let the universal set, $\xi$ = {$x : 1 \leq  x \leq  15$ and x is an  integer} set H = {x : x is a multiple of 3} and set K = {x : x is an even number}. Find $n(H' \cap K)$.

  1. $2$
  2. $5$
  3. $7$
  4. $13$
Reveal answer Fill a bubble to check yourself
B Correct answer
Explanation

H' includes all numbers from 1 to 15 that are not multiples of 3.


$H'\cap K ={2,4,8,10,14}$

$n(H'\cap K) =5$

Multiple choice mathematics and statistics binary operations properties of binary operations discrete mathematics sets and relations

Consider the following statements for non empty sets A, B and C
1 $\displaystyle A-\left ( B-C \right )=\left ( A-B \right )\cup C $
2 $\displaystyle A-\left ( B\cup C \right )=\left ( A-B \right )- C $
which of the statements given above is/are correct?

  1. 1 only

  2. 2 only

  3. Both 1 and 2

  4. Neither 1 nor 2
Reveal answer Fill a bubble to check yourself
B Correct answer
Explanation

1. $A- (B - C) = \displaystyle A-(B\cap C')$
$\displaystyle =A\cap (B\cap C')'$
$\displaystyle =A\cap (B'\cup (C'))$
$\displaystyle = A\cap (B'\cup C)$
Thus, $\displaystyle A-(B-C)\neq (A-B)\cup C$


2. A- $\displaystyle (B\cup C)=A\cap (B\cap C)'$
$\displaystyle = A\cap (B'\cap C)'$
$\displaystyle (A-B)-C=(A\cap B')-C$
$\displaystyle =A\cap B'\cap C'$
$\displaystyle \Rightarrow A-(B\cup C)=(A-B)-C$
Associative property. 

Multiple choice mathematics and statistics binary operations properties of binary operations discrete mathematics sets and relations

The set of integers $Z$ with the binary operation $*$ defined as $a * b = a + b+ 1$ for $a, b, Z$ is a group. The identity element of this group is

  1. $0$
  2. $1$
  3. $-1$
  4. $15$
Reveal answer Fill a bubble to check yourself
C Correct answer
Explanation

$a\ast b=a+b+1$   (a,b,z is a group)

at $a=-1 \Rightarrow a\ast b=-1+b+1=b$
at $b=-1  \Rightarrow a\ast b=a-1+1=a$
$\Rightarrow a\ast 0=a+0+1$
$\Rightarrow$ identity element is $-1$.

Multiple choice maths introduction to set cardinal number of a finite set cardinality of a set representation of sets

If $A\subset B$, then $n[P(A)]$ ______ $n[P(B)]$

  1. $=$
  2. $<$
  3. $\leq $
  4. $>$
Reveal answer Fill a bubble to check yourself
B Correct answer
Explanation

Assume $ A \subset B$ is true. 

Then, every element of $A$ i.e. $a _1,a _2, ... , a _n$ in A are also in B.

So, number of elements in $B$ will always be greater than no. of elements in $A$

And $P(A)$ will contain less number of subsets than $P(B)$

Hence, $n[P(A)] <  n[P(B)]$

Multiple choice maths introduction to set cardinal number of a finite set cardinality of a set representation of sets

Let $U$ be the universal set for sets $A$ and $B$ such that $n(A)=200 , n(B)=300$ and $n(A\cap B)=100$, then $n(A'\cap B')$ is equal to $300$ provided that $n(U)$ is equal to

  1. $600$
  2. $700$
  3. $800$
  4. $900$
Reveal answer Fill a bubble to check yourself
B Correct answer
Explanation

$n(A\cup B)=n(A)+n(B)-n(A\cap B)$
$=200+300-100$
$=400$
$n(A'\cap B')=n(A\cup B)'$
                    $=n(U) - n(A\cup B)$
$300=n(U)-400$
$n(U)=700$

Multiple choice maths introduction to set cardinal number of a finite set cardinality of a set representation of sets

Let $A$ and $B$ be two sets such that $\displaystyle n\left( A \right) =70$ and $\displaystyle n\left( B \right) =60$ and $\displaystyle n\left( A \cup B \right) =110 $. Then $\displaystyle n\left( A \cap B \right) $ is equal to

  1. $240$
  2. $20$
  3. $100$
  4. $120$
Reveal answer Fill a bubble to check yourself
B Correct answer
Explanation

Given $\displaystyle n\left( A \right) =70$ and $\displaystyle n\left( B \right) =60$ and $\displaystyle n\left( A\quad \cup \quad B \right) =110 $.

$\displaystyle n\left( A\quad \cup \quad B \right)=\displaystyle n\left( A \right)+n\left( B \right)-n\left( A\quad \cap \quad B \right)$

$\Rightarrow 110=70+60-\displaystyle n\left( A\quad \cap \quad B \right)$

$\therefore \displaystyle n\left( A\quad \cap \quad B \right)=20$

Hence, option B.