General Topic Doubt Set Theory & Algebra: Sets

Question

ϕ  ∈  PowerSet( A )

{ϕ } ⊆ PowerSet( A )
 

is both equal ? If not give explanation .

Comments

S={1,2,3,4,5}

Now when we talk about individual elements we say they belongs

like 2,3 ∈ S

when we talk about set of elements like A={4,5} we talk in terms of subset or superset

like here A ⊆ S means set S contains set A means   S ∩ A=A

NOw if someone asks like whether {1,4} ∈ S ? then if there is entry just like {1,4} then only we can say it belongs , and here in S no such element , so say statement is false.

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The empty set { }, denoted by ∅, is also a subset of any given set X. It is also always a proper subset of any set except itself.   

read here for basic concepts https://en.wikipedia.org/wiki/Set_theory

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Last question

is  { {{ } }} belong to S or subset of S equal ?

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Nope.

Phi means a single pair of curly braces with nothing in side it.

{ { { } } }

is equal to { { PHI } }.

Answer 1

Let S={1,2,3}

Find Subset of S = phi, {1},{2},{3}, {1,2}, {1,3}, {2,3},{1,2,3}.

Check few Question Here: 

1. phi ∈ S = in given set phi is not given So False.
2. phi ⊆ S  = phi is subset of every set So Ture
3. 1  ∈ S = 1 belongs to S So True
4. 2 ⊆  S = 2 ∈ S but 2 is not subset of S  So False. but {2} ⊆  S is True.

P(S) = {phi, {1},{2},{3}, {1,2}, {1,3}, {2,3},{1,2,3}} 

Check few Question Here: 

1. phi ∈ P(S) = in given set phi is given So True.
2. phi ⊆ P(S) = phi is subset of every set So Ture
3. 1  ∈ P(S) = 1 is not belongs to P(S).So False
4. 2 ⊆  P(S) = {2} ⊆  P(S) is False  but {{2}} ⊆  P(S) is Ture.

So what we can say :

x ∈ S means x element present in S

{x} ∈S means {x} belongs to S 

x ⊆ S always false since curly baces needed for sunset operation. except of x= phi

{x} ⊆ S means if x is present in S then only it is ture but if {x} is present and x is not present then it is not true.

Comments

if they given {2} subset of A

then on this basis can i say 2 belong to A .rt?

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yes ...i{2} is subset of A iff 2 belongs to A.