GATE CSE 2015 Set 1 | Question: 1‏6

Question

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?

1. $\varnothing \in 2^{A}$
2. $\varnothing  \subseteq 2^{A}$
3. $\left\{5,\left\{6\right\}\right\} \in 2^{A}$
4. $\left\{5,\left\{6\right\}\right\} \subseteq 2^{A}$

• A. I and III only
• B. II and III only
• C. I, II and III only
• D. I, II and IV only

Comments

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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Does  this phi in 2nd statement is  a set represents by { }  ?

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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. 

Answer 1

II. is trivially true. $\phi$ is a subset of any set.

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I. $ϕ∈2^A$. Means is $\phi$ an element of $2^A\ ?$ Yes, it is. $^{[1]}$

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III. Enumerate the elements of $2^A$. $\{5,\{6\}\}$ is an element in it. So, true.

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IV. Is $\{5,\{6\}\}$ a subset of $2^A\ ?$ This implicitly assumes that $\{5,\{6\}\}$ is a set. So, we can rephrase it as: are the elements $5$ and $\{6\}$ present in $2^A \ ?$

The answer is no.

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Option C

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$^{[1]}$ Power set is the set of all subsets.

Hence, the elements of a powerset are sets in themselves.

$\phi$ is the empty set (it literally is the definition of it), so $\phi$ will always belong $(∈)$ to any power set.

Answer 2

the power set (or powerset) of any set S is the set of all subsets of S, including the empty set and S itself.

 

A = {5, {6}, {7}}

Power set of A = 2^A = {Φ, {5}, {{6}}, {{7}}, {5, {6}}, {5, {7}}, {{6}, {7}}, {5, {6}, {7}}}

Statement I. Φ is element of power set of A. Therefore, Φ ϵ 2^A.

Statement II. Power set of A consists of all subsets of A and from the definition of a subset, ϕ is a subset of any set.

Therefore, Φ ⊆ 2^A Statement III {5, {6}} is element of power set of A. Therefore, {5, {6}} ϵ 2^A.

Statement IV {5, {6}} is element of power set of A. Therefore, {{5, {6}}} ⊆ 2^A.

Hence statement IV is false. Therefore option 3 is correct

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