An element of a set is never a subset of the set. For that the element must be inside a set

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For a set $A$, the power set of $A$ is denoted by $2^{A}$. If $A = \left\{5,\left\{6\right\}, \left\{7\right\}\right\}$, which of the following options are TRUE?

- $\varnothing \in 2^{A}$
- $\varnothing \subseteq 2^{A}$
- $\left\{5,\left\{6\right\}\right\} \in 2^{A}$
- $\left\{5,\left\{6\right\}\right\} \subseteq 2^{A}$

- I and III only
- II and III only
- I, II and III only
- I, II and IV only

Yes @Golam Murtuza

But why $\varnothing $ or $\left\{ \right\}$ is a subset of any set?

Set $A$ is subset of set $B$ iff $$ \forall x \left( x \in A \to x \in B \right)$$

Let $A=\varnothing $since no element can belong to empty set, $x \in A$ will be always $false$

Since $false \to anything$ is $true$ (Vacuos truth ) we can say $\varnothing $ is subset of any set.

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Power set of $A$ consists of all subsets of $A$ and from the definition of a subset, $\emptyset$ is a subset of any set. So, $\text{I}$ and $\text{II}$ are TRUE.

$5$ and $\{6\}$ are elements of $A$ and hence $\{5, \{6\} \}$ is a subset of $A$ and hence an element of $2^{A}$. An element of a set is never a subset of the set. For that the element must be inside a set- i.e., a singleton set containing the element is a subset of the set, but the element itself is not. Here, option $\text{IV}$ is false. To make $\text{IV}$ true we have to do as follows:

$\{5, \{6\} \}$ is an element of $2^{A}$. So, $\{ \{5, \{6\} \}\}\subseteq 2^{A}.$

So, option C.

$5$ and $\{6\}$ are elements of $A$ and hence $\{5, \{6\} \}$ is a subset of $A$ and hence an element of $2^{A}$. An element of a set is never a subset of the set. For that the element must be inside a set- i.e., a singleton set containing the element is a subset of the set, but the element itself is not. Here, option $\text{IV}$ is false. To make $\text{IV}$ true we have to do as follows:

$\{5, \{6\} \}$ is an element of $2^{A}$. So, $\{ \{5, \{6\} \}\}\subseteq 2^{A}.$

So, option C.

21 votes

We use "subset" symbol to compare 2 sets ...Ex: Set A is a subset of set B iff every elements of set A is in set B.

We use "belongs to" symbol to compare a set and an element. Ex: Whether an element is present inside a set or not.

{5,{6}} is not a subset of 2^{A} .. Because 5 is not present in 2^{A}. {5} is actually present in 2^{A}.

{5} (a set containing an element 5) is different from 5 (an element)...