Tag: intervals

Questions Related to intervals

The number of subsets of the set $A={ { a } _{ 1 },{ a } _{ 2 },.........{ a } _{ n }} $ which contain even number of elements is

maths introduction to set intervals subsets subsets and supersets

  1. ${ 2 }^{ n-1 }$

  2. ${ 2 }^{ n }-1$

  3. ${ 2 }^{ n }-2$

  4. ${ 2 }^{ n }$

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Correct Option: A
Explanation:
The total no of subsets of $A$ is the cardinality of the power set of $A$. 

So, if $|A|=n$ then $|P(A)|=2^n$. 

Therefore total no of subsets of $A$ is $2^n$.

Similarly,

The even number of events is given by, $2^{n-1}$

Let $A = {a, b, c}, B = {a}, C = {a, b}$ then,  which set is the superset of $C$? 

maths set concepts intervals subsets subsets and supersets

  1. Set $A$

  2. Set $B$

  3. Set $A$ and Set $B$

  4. None of these

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Correct Option: A
Explanation:

$\text{Clearly, set A contain all elements of set C}$
$\Rightarrow \text{A is superset of C}$

If U = {1, 2, 3, .......}; A = {2, 4, 6, 8, .......}; B = {1, 3, 5, .......}, then find (A $\cup$ B)'.

maths set concepts intervals subsets subsets and supersets

  1. A'

  2. B

  3. A

  4. $\phi$

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Correct Option: D
Explanation:
Given,
$U=(1, 2, 3,....)$
$A=\{2, 4, 6, 8,...\}$
$B=\{1, 3, 5, 7,....\}$

We know,
$(A\cup B)'=U-(A\cup B)$

here
$A\cup B=\{1, 2, 3,....\}=U$

so $(A\cup B)'=\phi$.

The number of subsets with two elements, of the set $S+{1,2,3,4,.....,10}$ such that minimum of the two numbers is less than $6$ is 

maths set concepts intervals subsets subsets and supersets

  1. $35$

  2. $38$

  3. $30$

  4. $40$

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Correct Option: A

Which of the following sets is a universal set for the other four sets? 

(a) The set of even natural numbers 

(b) The set of odd natural numbers

(c) The set of natural numbers 

(d) The set of negative numbers 

(e) The set of integers 

maths set concepts intervals subsets subsets and supersets

  1. $(e)$

  2. $(a)$

  3. $(b)$

  4. $(c)$

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Correct Option: A
Explanation:

We know that $\mathbb{N} \subset \mathbb{Z}$ where $\mathbb{N}$ represents set of natural numbers and $\mathbb{Z}$ represents set of all positive and negative integers.

Since $\mathbb{N} \subset \mathbb{Z}$,

                                 $(a),(b),(c)$ are subsets of $(e)$        $...(1)$

Since $\mathbb{Z}$ represents set of all positive and negative integers.

                                      $(d)$ is a subset of $(e)$                   $...(2)$

From $(1)$ and $(2)$ we get

$(a),(b),(c),(d)$ are subsets of $(e)$.

Hence $(e)$ is the universal set for the other four sets.

State whether the following statement is true or false
 $0$ $\epsilon$ $\phi$

maths set concepts intervals subsets subsets and supersets

  1. True

  2. False

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Correct Option: B
Explanation:

We have to state whether the statement "$0 \in \phi$" is true or false.

We know that, $\phi$ represents null set.

The null set, also called the empty set, is the set that does not contain any element.

Thus by the definition of null set, $0 \notin \phi$

Hence the given statement is false.