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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?

1. $\phi \in 2^{A}$
2. $\phi \subseteq 2^{A}$
3. $\left\{5,\left\{6\right\}\right\} \in 2^{A}$
4. $\left\{5,\left\{6\right\}\right\} \subseteq 2^{A}$
1. I and III only
2. II and III only
3. I, II and III only
4. I, II and IV only

edited | 5.4k views
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why {5,{6}}⊆2A  is incoorect

+9
$\{5, \{6\}\} \in 2^A$
So, $\{\{5, \{6\}\}\} \subseteq 2^A$
$\{5, \{6\}\} \not\subseteq 2^A$
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IV is false, {5, {6}} is not a subset, but {{5, {6}}} is.

What does this mean?
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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 { }  ?

Power set of $A$ consists of all subsets of $A$ and from the definition of a subset, $\phi$ is a subset of any set. So, $I$ and $II$ 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 $IV$ is false. To make 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.
by Veteran (431k points)
edited by
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Why not option D? Can you please explain why not?
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If A and B are sets and if every element of A is an element of B, we say that A is a subset of B, or B includes A, and we write:

AB

So what I'm missing here? Why not IV is correct

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I have modified the explanation. Hope it is clear now :)
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Just for a pair of brackets!! Thanks.
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Shouldnt $\phi$ to be a subset be represented as $\{ \phi \}$ ?? In option II
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Well, that's a great doubt actually- don't think it is a stupid doubt.

As I told ∅ is a subset of any set. And that is why it is a member of power set.

As you told {∅} thus becomes a subset of the power set of A.

But power set is also a set. And by definition of subset, ∅ is a subset of this set also.
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So the definition makes it so.
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yes. Definition is the key.

http://en.wikipedia.org/wiki/Empty_set

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But I've just seen an example in Tremblay-Manohar book which contradicts your answers.(Correct me If i'm wrong). Example goes as follows :
A={ {1}, 2, 3 }

{ {1} ,2 } ⊆ A

So IV should be correct. Right?
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subset of A is correct. But IV option says subset of power set of A which is wrong.
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I've gone through the definition again and the concept is pretty clear now. Thanks.
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@Surya_Teja Sir I am having trouble in understanding option IV

5, {6}$\in$ A

{ 5, {6} } ⊆ A

{ 5, {6} } $\in$  PowerSet( A )

{ { 5, {6} } } ⊆ PowerSet( A )

Similarly

$\phi \notin$ A

$\phi$ ⊆ A

$\phi$  $\in$  PowerSet( A )

{$\phi$ } ⊆ PowerSet( A )

@Arjun Sir,
Is my understanding Correct ?
+1
yes. You are right..
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@pC....in the last line Bracket should not be present.

It should be like   ϕ  ⊆ PowerSet( A )
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@vaishali both {phi} and phi are subset of powerset(A) because:

1. Phi is subset of every set

2. Powerset(A) contains phi as one of its element so {phi} is also a subset of powerset(A), just like how {{5, {6}}} is a subset of powerset(A)
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#Arjun Sir i think u have missed a Pair of {} ... it should be { {5, {6} } } ⊆ 2A ......

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Thank you :)
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Option B becomes clear when we look at the definition of subset. A set $A$ is the subset of $B$ if for all elements $x$ , if $x \in A \implies x \in B$.

Now since the set $\emptyset$ has no elements, then the sentence becomes a vacuous truth and therefore, is true in nature.
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@Arjun sir

$\phi \subset 2^{A}$ or $\phi \subseteq 2^{A}$ ?? m consfused in this

is $\phi$ a proper OR improper subset of Non empty set ?

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@jatin khachane 1

Take any set, If you get a subset by excluding all (phi) or by including all (set itself), then it is not proper subset.

For proper subset for all x(If a belongs to subset => it belongs to Set and there exist an element which is in Set but not in subset).

Hope it is clear now.

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so phi will be proper subset of any non empty set as, all element of phi are present in any non empty set and there exist some element which is in that non empty set but not in phi ..right

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+1
Can we say ϕ is subset of every set and ϕ ∈ to every powerset. ??
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yes.
+4

This slide from CS103 of Stanford will clear the doubts that everyone is having. https://web.stanford.edu/class/archive/cs/cs103/cs103.1176/notes/Guide%20to%20Elements%20and%20Subsets.pdf

The whole CS 103 is recommended for everyone.

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@Arjun Sir , For given question

phi is Subset of powerset of a. --This Phi is of the Power set itself.

Also {phi} -- This phi is a set of phi which is coming from A ..

So {phi} is also subset of Power set of A

Right???

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

2A  = p = {  ϕ, {5}, {{6}}, {{7}}, {5, {6}}, {5, {7}}, {{6}, {7}}, {5, {6}, {7}}  }

by Active (1.5k points)
edited by

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 2A .. Because 5 is not present in 2A. {5} is actually present in 2A

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

by Loyal (8k points)

option C

phi is subset of every set.

phi is the first element of the power set

{5,{6}} ⊆ A

by Active (1.2k points)
edited by