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 relations cartesian product of sets cartesian product of two sets cartesian product

If $A=\left {1, 2,3\right }$ and $B=\left {3,8\right }$, then $(A\cup B)\times (A\cap B)$ is equal to

  1. $\left \{(8,3), (8,2), (8,1), (8,8)\right \}$
  2. $\left \{(1,2), (2,2), (3,3), (8,8)\right \}$
  3. $\left \{(3,1), (3,2), (3,3), (3,8)\right \}$
  4. $\left \{(1,3), (2,3), (3,3), (8,3)\right \}$
Reveal answer Fill a bubble to check yourself
D Correct answer
Explanation

Given, $A=\left {1, 2,3\right }$ and $B=\left {3,8\right }$,
Therefore, $A\cup B=\left {1,2,3\right }\cup \left {3,8\right }=\left {1,2,3,8\right }$
and $A\cap B=\left {1,2,3\right }\cap \left {3,8\right }=\left {3\right }$
$\therefore (A\cup B)\times (A\cap B)=\left {1,2,3,8\right }\times \left {3\right }$
$=\left {(1,3), (2,3), (3,3), (8,3)\right }$

Multiple choice mathematics and statistics relations cartesian product of sets cartesian product of two sets cartesian product

What is the Cartesian product of $A = \left {1, 2\right }$ and $B = \left {a, b\right }$?

  1. $\left \{(1, a), (1, b), (2, a), (b, b)\right \}$
  2. $\left \{(1, 1), (2, 2), (a, a), (b, b)\right \}$
  3. $\left \{(1, a), (2, a), (1, b), (2, b)\right \}$
  4. $\left \{(1, 1), (a, a), (2, a), (1, b)\right \}$
Reveal answer Fill a bubble to check yourself
C Correct answer
Explanation

If $A $ and $B$ are two non empty sets, then the Cartesian product $A \times B$ is set of all ordered pairs $(a,b)$ such that $a\in A$ and $b\in B$.


Given $A ={1,2}$ and $B = {a,b}$

Hence $A\times B = {(1,a),(1,b),(2,a),(2,b)}$ 

Multiple choice mathematics and statistics relations cartesian product of sets cartesian product of two sets cartesian product

If A and B are two non-empty sets having n elements in common, then what is the number of common elements in the sets $A\times B$ and $B\times A$?

  1. $n$
  2. $n^2$
  3. $2n$

  4. Zero
Reveal answer Fill a bubble to check yourself
B Correct answer
Explanation

Say A has x elements and B has y elements in total. 


Their cartesian product $A\times B$ will have $x\times y$ elements.

Hence if they have n elements in common. $n^2$ common elements are present in the products $A\times B$ and $B\times A$

Multiple choice mathematics and statistics relations cartesian product of sets cartesian product of two sets cartesian product

Let $A=\left{ x\in W,the\quad set\quad of\quad whole\quad numbers\quad and\quad x<3 \right} $

$B=\left{ x\in N,the\quad set\quad of\quad natural\quad numbers\quad and\quad 2\le x<4 \right} $ and $C=\left{ 3,4 \right} $, then how many elements will $\left( A\cup B \right) \times C$ conatin?

  1. $6$
  2. $8$
  3. $10$
  4. $12$
Reveal answer Fill a bubble to check yourself
B Correct answer
Explanation

$A={0,1,2}, B={2,3}$ and $C={3,4}$
$A \cup B={0,1,2,3}$
No. of elements in $(A\cup B)\times C$ $=$ No. of elements in $(A \cup B)\times $ No. of elements in $C$$=4\times 2=8$

Multiple choice mathematics and statistics relations cartesian product of sets cartesian product of two sets cartesian product

Let $A = \left{ a,b,c,d \right}$ and $ B=\left{ x,y,z \right}$. What is the number of elements in $ A\times B$?

  1. $6$
  2. $7$
  3. $12$
  4. $64$
Reveal answer Fill a bubble to check yourself
C Correct answer
Explanation

Given sets are $A={a,b,c,d}$ and ${x,y,z}$ 

So $A\times B={(a,x),(a,y),(a,z),(b,x),(b,y),(b,z),(c,x),(c,y),(c,z),(d,x),(d,y),(d,z)}$
No. of elements in $A\times B$ is $3\times 4=12$

Multiple choice mathematics and statistics relations cartesian product of sets cartesian product of two sets cartesian product

If $A = \left{ 1,2 \right}$, $B = \left{ 2,3 \right}$ and $ C = \left{ 3,4 \right}$, then what is the cardinality of $ \left( A\times B \right) \cap \left( A\times C \right) $

  1. $8$
  2. $6$
  3. $2$
  4. $1$
Reveal answer Fill a bubble to check yourself
C Correct answer
Explanation
$A = \left\{ 1,2 \right\}$, $B = \left\{ 2,3 \right\}$ and $ C = \left\{ 3,4 \right\}$

Now, $A\times B=\{ (1,2),(1,3),(2,2),(2,3)\}$ 

$A\times C=\{ (1,3),(1,4),(2,3),(2,4)\} $ And

$ (A\times B)\cap (A\times C)=\{ (1,3),(2,3)\} $

So cardinality is $2$.

Hence, option C is correct.

Multiple choice mathematics and statistics relations cartesian product of sets cartesian product of two sets cartesian product

If two sets $A$ and $B$ are having $39$ elements in common, then the number of elements common to each of the sets $A\times B$ and $B\times A$ are

  1. ${ 2 }^{ 39 }$
  2. ${ 39 }^{ 2 }$
  3. $78$
  4. $351$
Reveal answer Fill a bubble to check yourself
B Correct answer
Explanation

If set $A$ and set $B$ have $39$ common elements, then the number of common elements in set $A\times B$ and set $B\times A\,=39^2$

Multiple choice mathematics and statistics relations cartesian product of sets cartesian product of two sets cartesian product
Suppose $S=\{1,2\}$  and $T=\{a,b\}$  then $T \times S$
  1. <span class="math">$(a,1),(a,2),(b,1),(b,2)$
    <span class="MJX_Assistive_MathML">B×A={(a,1),(a,2),(b,1),(b,2)}
  2. $(1,a),(2,b),(b,1),(b,2)$
  3. $(a,1),(a,2),(1,b),(2,b)$

  4. None of the above
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation
Given : $S=\{ 1,2\} ,T=\{ a,b\} $
$T\times S$ is the set of ordered pair of elements of $T$ and $S$ respectively
Then, $T\times S=\{a,b\}\times \{1,2\}$
$T\times S=\{ (a,1),(a,2),(b,1),(b,2)\} $
Multiple choice mathematics and statistics relations cartesian product of sets cartesian product of two sets cartesian product

$\left (A \cap B  \right ) \times C$

  1. $\left (A \times B \right ) \cap \left (B \times C \right )$
  2. $\left (A \times C \right ) \cap \left (B \times C \right )$
  3. $\left (A \times B \right ) \cup \left (B \times C \right )$
  4. $\left (A \times B \right ) \cup \left (A \times C \right )$
Reveal answer Fill a bubble to check yourself
B Correct answer
Explanation

The Cartesian product distributes over intersection: (A intersect B) x C = (A x C) intersect (B x C).