Tag: introduction to set

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Questions Related to introduction to set

Multiple choice mathematics and statistics introduction to set union and intersections union and intersection of sets basic operations on sets

A is a set containing $n$ elements. $A$ subset $P$ of $A$ is chosen. the set $A$ is reconstructed by replacing the elements of $P.A$ subset $Q$ of $A$ is again chosen. the number of ways of choosing $P$ and $Q$ so that $P \cap Q$

  1. $9. ^{n}C _{2}$
  2. $3^{n}- ^{n}C _{2}$
  3. $^{n}C _{2}.3^{n-2}$
  4. $4^{n}-3^{n}$
Reveal answer Fill a bubble to check yourself
A Correct answer
Multiple choice mathematics and statistics introduction to set union and intersections union and intersection of sets basic operations on sets

Let $A=\left{ a,b,c,d \right} ,B=\left{ b,c,d,e \right}$. Then $n\left[ \left( A\times B \right) \cap \left( B\times A \right)  \right]$ is equal to 

  1. $3$
  2. $6$
  3. $9$
  4. $none$
Reveal answer Fill a bubble to check yourself
C Correct answer
Explanation

The number of elements in the Cartesian product of two sets is the product of their cardinalities. Since both sets A and B have 4 elements, the intersection of A cross B and B cross A consists of the ordered pairs (x, y) where both x and y belong to the intersection of A and B. The intersection of A and B contains 3 elements, so the number of such pairs is 3 squared, which is 9.

Multiple choice mathematics and statistics introduction to set union and intersections union and intersection of sets basic operations on sets

The set of all points where the function $f(x)=||x|$ is twice differentiable is

  1. $(-\infty, \infty)$
  2. $(-\infty, 0)\cup (0, \infty)$
  3. $(0, \infty)$
  4. $[0, \infty)$
Reveal answer Fill a bubble to check yourself
B Correct answer
Explanation

The function can be simplified based on the absolute value definition. For x different from zero, the function equals x squared or negative x squared, which is infinitely differentiable. At x equals zero, the first derivative exists but the second derivative does not exist, so the function is twice differentiable everywhere except at zero.

Multiple choice mathematics and statistics introduction to set union and intersections union and intersection of sets basic operations on sets

Let $S=\left{ \left( x,y \right) :\dfrac { y\left( 3x-1 \right)  }{ x\left( 3x-2 \right)  } <0 \right}$ and $S'=\left{ \left( x,y \right) \in A\times B;\ -1\le A\le 1,-1\le B\le 1 \right} $ There area of $S\cap S'$ is

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

The region S is defined by the inequality involving fractions, and S prime is a bounded square region. Solving the inequality for x and y and intersecting with the square yields a region of area 2. Careful sketching of the sign scheme for the rational expression reveals the valid domain.

Multiple choice mathematics and statistics introduction to set union and intersections union and intersection of sets basic operations on sets

Let A={1, 2, 3, 4), B={2, 3, 4, 5}, then $n{ (A\times B)\cap (B\times A)} =$?

  1. 13

  2. 16

  3. 9

  4. 10
Reveal answer Fill a bubble to check yourself
C Correct answer
Explanation

The sets A and B share three common elements, namely 2, 3, and 4. The intersection of A cross B and B cross A is equivalent to the Cartesian product of the intersection of A and B with itself. Since the intersection has 3 elements, its square has 3 times 3, or 9 elements.

Multiple choice mathematics and statistics introduction to set union and intersections union and intersection of sets basic operations on sets

Let $P={ \theta :sin\theta -cos\theta =\sqrt { 2 } cos\theta } $ and $Q={ sin\theta + cos\theta =\sqrt { 2 } sin\theta } $ be two sets. Then:

  1. $P\subset Q\quad and\quad Q-P\neq \emptyset $
  2. $Q\subset P$
  3. $P\subset Q$
  4. $P=Q$
Reveal answer Fill a bubble to check yourself
D Correct answer
Explanation

$P = \left{ {\theta :\sin \theta  - \cos \theta  = \sqrt 2 \cos \theta } \right}$

$Q = \left{ {\theta :\sin \theta  + \cos \theta  = \sqrt 2 \sin \theta } \right}$
From $P$
$\sin \theta  - \cos \theta  = \sqrt 2 \cos \theta $
$\sin \theta  = \left( {\sqrt 2  + 1} \right)\cos \theta $
$\frac{{\sin \theta }}{{\cos \theta }} = \left( {\sqrt 2  + 1} \right)$
$\tan \theta  = \left( {\sqrt 2  + 1} \right)$
from $Q$
$\sin \theta  + \cos \theta  = \sqrt 2 \sin \theta $
$\sin \theta \left( {\sqrt 2  - 1} \right) = \cos \theta $
$\frac{{\sin \theta }}{{\cos \theta }} = \left( {\sqrt 2  - 1} \right)$
$\tan \theta  = \left( {\sqrt 2  - 1} \right)$
$\tan \theta  = \frac{1}{{\sqrt 2  - 1}} \times \frac{{\sqrt 2  + 1}}{{\sqrt 2  + 1}} = \sqrt 2  + 1$
$\therefore P = Q$
Hence,
option $(D)$ is correct answer.

Multiple choice mathematics and statistics introduction to set union and intersections union and intersection of sets basic operations on sets

If $A = {1, 2, 3, 4, 5}, B = {2, 4, 6, 8}$ and C= ${3,4,5,6}$, 

then verify : $A - (B \cup C) = (A - B) \cap (A - C)$.

  1. True

  2. False
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation
Given, $A = \{1, 2, 3, 4, 5\}, B = \{2, 4, 6, 8\}$ and $C=\{3,4,5,6\}$

For the LHS:

Union of two sets will have the elements of both sets.

So, $ B \cup C = \{2,3,4,5,6,8 \}$ 

$ A - (B \cup C) $ will have elements of $A$ which are not in $ (B \cup C) $

So, $ A - (B \cup C) = \{ 1 \}$ ..... $(1)$

For the RHS:

$ A - B $ will have elements of $A$ which are not in $B$.

So, $ A - B = \{ 1,3,5 \}$  

$ A - C $ will have elements of $A$ which are not in $C$.

So, $ A - C = \{ 1,2 \}$  

Intersection of two sets has the common elements of both the sets. 

$\Rightarrow (A - B) \cap (A - C) = \{1\}$ ..... $(2)$

From $(1)$ and $(2),$ we have

$ A - (B \cup C) =(A - B) \cap (A - C) $

Hence, the given expression is true.
Multiple choice mathematics and statistics introduction to set union and intersections union and intersection of sets basic operations on sets

Let $A = {$ multiples of $3$ less than $20 }$
      $B = {$ multiples of $5$ less than $20}$
Then  $A$ $\displaystyle\cap$ $B$ is

  1. $\{3, 5\}$
  2. $\{5, 9\}$
  3. ${15}$
  4. $\displaystyle\phi $
Reveal answer Fill a bubble to check yourself
C Correct answer
Explanation

$A = {$ multiples of $ 3 $ less than $20}$

    $= {3,6,9,12,15,18}$
$B={ $ multiples of $5$ less than $20}$
    $= {5,10,15}$

$A \cap B = 15$

Multiple choice mathematics and statistics introduction to set union and intersections union and intersection of sets basic operations on sets

Let $P _1$ be the set of all prime numbers, i.e., $P _1=\left {2, 3, 5, 7, 11, ....\right }$, Let $Pn=\left {np|p\epsilon P _1|\right }$, i.e., the set of all prime multiples of n. Then which of the following sets is non empty?

  1. $P _1\cap P _{23}$
  2. $P _7\cap P _{21}$
  3. $P _{12}\cap P _{20}$
  4. $P _{20}\cap P _{24}$
Reveal answer Fill a bubble to check yourself
C Correct answer
Explanation

Check by option
$P _{12}=\left {24, 36, 60, 84, ....\right }$
$P _{20}=\left {40, 60, 100, .....\right }$
$P _{12}\cap P _{20}$ has common element.