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

If $A = \left {1, 2, 3, 4, 5, 6, 7, 8\right }$ and $B \left {1, 3, 5, 7\right }$, then find $A - B$ and $A \cap B$

  1. $\left \{3, 5\right \}$ and $\left \{2, 4, 6\right \}$
  2. $\left \{2, 4, 6\right \}$ and $\left \{1, 5\right \}$
  3. $\left \{2, 4, 6, 8\right \}$ and $\left \{1, 3, 5, 7\right \}$
  4. $\left \{1, 3, 5, 8\right \}$
Reveal answer Fill a bubble to check yourself
C Correct answer
Explanation

$A=\{1,2,3,4,5,6,7,8\}$

$B=\{1,3,5,7\}$

$A-B=\{1,2,3,4,5,6,7,8\} - \{1,3,5,7\} = \{2,4,6,8\}$

$A \cap B = \{1,2,3,4,5,6,7,8\} \cap \{1,3,5,7\} =\{1,3,5,7\}$

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

In a certain group of $36$ people, $18$ are wearing hats and $24$ are wearing sweaters. If six people are wearing neither a hat nor a sweater, then how many people are wearing both a hat and a sweater?

  1. $30$
  2. $22$
  3. $12$
  4. $8$
Reveal answer Fill a bubble to check yourself
C Correct answer
Explanation

$n(U)=36$

$n(Hats) =18 =n(H) $

$n(Sweaters) =24 =n(S) $

n(Wearing neither hat nor Sweater) =6

$n(S \cup H) = n(U)-$ n(Wearing neither hat nor sweater) $= 36-6 = 30 $

$n(S \cup H) = n(S) + n(H)-n(S\cap H) $

$30 = 24+18-n(S\cap H) $

$n(S\cap H)=42-30 = 12 $

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

In a class of $80$ children, $35$% children can play only cricket, $45$% children can play only table-tennis and the remaining children can play both the games. In all, how many children can play cricket?

  1. $55$
  2. $44$
  3. $36$
  4. $28$
Reveal answer Fill a bubble to check yourself
B Correct answer
Explanation

n(Children who can play only cricket) $ = 35\% = 80\times \dfrac{35}{100} = 28$

n(Children who can play only table tennis) $ = 45\% = 80\times \dfrac{45}{100} = 36$

$n(C \cap T) = 80-28-36 = 16$

n(Children can play cricket) $= 28+16 =44$

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

If x belongs to set of integers, A is the solution set of $2(x-1)< 3x-1$ and B is the solution set of $4x-3\leq 8+x$, find A$\cap$B.

  1. $\left\{0, 1, 2\right\}$
  2. $\left\{1, 2, 3\right\}$
  3. $\left\{0, 1, 2, 3\right\}$
  4. $\left\{0, 2, 4\right\}$
Reveal answer Fill a bubble to check yourself
C Correct answer
Explanation
$2(x-1)< 3x-1$
$\Rightarrow 2x-2< 3x-1$
$\Rightarrow 2x-3x< -1+2$
$\Rightarrow -x< 1$
$\Rightarrow x> -1$

$4x-3\leq 8+x$
$4x-3-x+3\leq 8+x-x+3$
$\Rightarrow 4x-x\leq 8+3$
$\Rightarrow 3x\leq 11$
$x\leq \dfrac{11}{3}\Rightarrow x\le 3.67$
Then $-1< x\leq 3.67$
Then, $A\cap B$ is $ \{0,1,2,3 \}$
Multiple choice mathematics and statistics introduction to set union and intersections union and intersection of sets basic operations on sets

If $A = \left {2, 3, 4, 8, 10\right }, B = \left (3, 4, 5, 10, 12\right }, C = \left {4, 5, 6, 12, 14\right }$, then $(A\cap B)\cup (A\cap C)$ is equal to

  1. $\left \{3, 4, 10\right \}$
  2. $\left \{2, 9, 10\right \}$
  3. $\left \{4, 5, 6\right \}$
  4. $\left \{3, 5, 14\right \}$
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

$ A={2,3,4,8,10} , B={3,4,5,10,12} , C={4,5,6,12,14}$


To find : $(A\cap B)$ $\cup  (A\cap C)$

$A\cap B$ = elements common in sets $A$ and $B$. 
$A\cap C$ = elements common in sets $A$ and $C$ 

$A\cap C = \{4\}$ 
$(A\cap B)\cup (A\cap C)=$elements in $(A\cap B)$ and $(A\cap C)$
$(A\cap B) \cup  (A\cap C) = \{3,4,10\} \cup \{4\} $
$(A\cap B) \cup  (A\cap C)={3,4,10}$

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

Which is the simplified representation of 
$\left( {{A^/} \cap \,{B^/} \cap \,C} \right) \cup \left( {B\, \cap \,C} \right) \cup \left( {A \cap \,C} \right)$  

where A,B,C are subsets of X

  1. A

  2. B

  3. C
  4. $X \cap \,\left( {A \cup B \cup C} \right)$
Reveal answer Fill a bubble to check yourself
C Correct answer
Explanation

Using set theory laws, simplify the expression ((A' intersect B') intersect C) union (B intersect C) union (A intersect C). Notice that C is a common factor across the union terms: (A' intersect B') intersect C can be written as (A union B)' intersect C. Distributing C across the union with other terms using set identities yields simply C.

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

If $aN=\left{ ax:x\epsilon N \right}$, then the set $3N\cap 7N$ is

  1. $21\ N$
  2. $10\ N$
  3. $4\ N$

  4. None of these
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

The set nN represents multiples of n. The intersection of multiples of 3 (3N) and multiples of 7 (7N) consists of multiples of the least common multiple of 3 and 7, which is 21. Thus, 3N intersect 7N = 21N.