Tag: union and intersection of sets

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

Let $n$ be a fixed positive integer. Define a relation $R$ on $I$ (the set of all integers) as follows: a R b iff $n|(a-b)$ i.e., iff (a-b) is divisible by n. Show that $R$ is an equivalence relation on 1.

  1. $R$ is an equivalence relation on 1.
  2. $R$ is not an equivalence relation on 1.
  3. $R$ is a symjetric relation on 1.
  4. $R$ is an identity relation on 1.
Reveal answer Fill a bubble to check yourself
A,C Correct answer
Explanation

If $A\subseteq B$ 
$\therefore A\cap B=A$ 
R is reflexive since for any integer $a$ we have $a-a=0$ and $0$ is divisible by $n$.
Hence $aRa\quad \forall a\in I$

R is symmetric, $aRb$. Then by definition of $R$, $a-b=nk$ where $k\in I$.
Hence $b-a=\left( -k \right) n$ where $-k\in I$ and so $bRa$.
Thus we shown that $aRb\Rightarrow bRa$

R is transitive, let $aRb$ and $bRc$. then by definition of $R$, we have
$a-b={ k } _{ 1 }n$ and $b-a={ nk } _{ 2 }$
where ${ k } _{ 1 },{ k } _{ 2 }\in I$
It follow that $a-c=\left( a-b \right) +\left( b-c \right) ={ k } _{ 1 }n+{ k } _{ 2 }n=\left( { k } _{ 1 }+{ k } _{ 2 } \right) n$ 

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

A and B are two sets such that $A\displaystyle\cup B$ has $18$ elements If A has $8$ elements and B has $15$ elements then the number of elements in $A\displaystyle\cap  B$ will be: 

  1. $5$
  2. $8$
  3. $7$
  4. $4$
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

$n(A \cup B)=n(A)+n(B)-n(A \cap B)$

$n(A \cap B) = n(A)+n(B)-n(A \cup B)=8+15-18=5$

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

Let A = { even number} B = {prime numbers} Then A $\displaystyle\cap $ B equals: 

  1. {odd number}

  2. {composite number}

  3. {2}

  4. {whole numbers}
Reveal answer Fill a bubble to check yourself
C Correct answer
Explanation

Given: A = ${2, 4, 6, ...}$
     B = ${2, 3, 5, ...}$
$\displaystyle \therefore $ 2 is the only even prime number $\displaystyle A\cap B=\left { 2 \right }$

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

Let $A = { x | x$ $\displaystyle \in $ $N$, $x$ is a multiple of 2$ }$
     $ B = { x | x$ $\displaystyle \in $ $N$, $x$ is a multiple of 5$}$
     $C = {x | x$ $\displaystyle \in $ $N$, $x$ is a multiple of 10$}$
The set $\displaystyle\left ( A\cap B \right )\cap C$ is equal to:

  1. $A$
  2. $\displaystyle A \cap C$
  3. $B$
  4. $C$
Reveal answer Fill a bubble to check yourself
D Correct answer
Explanation
Given A = $\{ 2, 4, 6, 8, 10, 12, 14,...\}$  
B = $\{5, 10, 15, 20, 25,...\}$
C = $\{10, 20, 30, 40, ...\}$
$\displaystyle \Rightarrow $$\displaystyle A\cap B$ = $\{ 10, 20, 30, ...\}$ 
($\displaystyle A\cap B$) $\displaystyle \cap C=$ $\{10, 20, 30, ...\}$ = C
Multiple choice mathematics and statistics introduction to set union and intersections union and intersection of sets basic operations on sets

There are $19$ hockey players in a club. On a particular day $14$ were wearing the prescribed hockey shirts while $11$ were wearing the prescribed hockey paints. None of them was without a hockey pant or a hockey shirt. How many of them were in complete hockey uniform ?

  1. $8$
  2. $6$
  3. $9$
  4. $7$
Reveal answer Fill a bubble to check yourself
B Correct answer
Explanation

Let $P$ and $S$ represents the sets of hockey player wearing the prescribed hockey pants and shirts respectively.
Then $n(P) = 11$, $n(S) = 14$, $n$$\displaystyle \left ( P\cup S \right )$ $= 19$
$\displaystyle \Rightarrow $ $\displaystyle \left ( P\cap S \right )$ $=$ no of people wearing both pantand shirt
$= n(P) + n(S) - n$$\displaystyle \left ( P\cup S \right )$
$= 11 + 14 - 19 = 6$

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

If $\displaystyle A\cap B=A$ and $\displaystyle B\cap C=B$ then $\displaystyle A\cap C$ is equal to :

  1. $B$
  2. $C$
  3. $\displaystyle B\cup C$
  4. $A$
Reveal answer Fill a bubble to check yourself
D Correct answer
Explanation

Given:-

$A\cap B=A$ and $B\cap C$
So,$A$ is subset of $B$.
B is a subset of C.Since $B\cap C =B$
$A$ is a subset of $B$ and $B$ is subset of $C$.
So, $A$ and $B$ is subset of $C$.
So, $A\cap C=A$

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

In a group of $500$ people $200$ can speck Hindi alone while only $125$ can speck English alone The number of people can speck both Hindi and English is

  1. $175$
  2. $325$
  3. $300$
  4. $375$
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

Total no. of people$=500$

People who speak Hindi only$=200$
People who speak English only$=125$
$\therefore$ The number of people who can speak both Hindi and English$=500-(200+125)$
$\Rightarrow 500-325=175$

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

Given $A={a,b,c,d,e,f,g,h}$ and $B={a,e,i,o,u}$ then $A\cap B$ is equal to

  1. $\{a,e\}$
  2. $\{f,g\}$
  3. $\{g,h\}$
  4. $\{i,u\}$
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation
Given, $A=\{a,b,c,d,e,f,g,h\}$ and $B=\{a,e,i,o,u\}$

$ A$ intersection $B $, which means a new set can be constructed by determining which members are common among the two sets.

So as per the question:-
$A\cap B=\left\{ a,e \right\}$