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Set Theory (Posted on 2014-04-19) Difficulty: 3 of 5

  
The following is a list of 15 thought provoking statements
about sets where a true or false answer is needed.
A denotes any set, F the set {1,2}, φ the empty set,
and ℘(S) the set of all subsets of set S (i.e, the power
set of S).

a)   φ∩{φ} = {φ}
b)   φ∪{φ} = {φ,{φ}}
c)   φ ∈ ℘({φ,{φ}})
d)   {φ} ⊆ A
e)   φ ⊆ A
f)   φ ⊆ ℘(A)
g)   {{φ}} ⊆ ℘(φ)
h)   {φ}∩φ = φ
i)   ℘(φ) = {φ,{φ}}
j)   φ ∈ A
k)   φ ∈ ℘(A)
l)   {φ} ∈ ℘(A)
m)   {φ}∪φ = {φ}
n)   φ ⊆ ℘(F)-φ
o)   {φ} ⊆ {{φ,{φ},{{φ}}}}

From "Bridge to Abstract Mathematics",1987
          by Ronald P. Morash.
  

  Submitted by Bractals    
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Solution: (Hide)

  
A denotes any set, F the set {1,2}, φ the empty set,
and ℘(S) the set of all subsets of set S (i.e, the power
set of S).To reduce the number of {} we will denote
{φ} by θ
     STATEMENT        (T/F)     REASON
     ---------        -----     ------
a)   φ∩θ = θ   		F   	φ∩θ = φ ≠ θ
b)   φ∪θ = {φ,θ}       	F  	φ∪θ = θ ≠ {φ,θ}
c)   φ ∈ ℘({φ,θ})     	T       φ ∈ ℘(A) ∀A
d)   θ ⊆ A            	F      	θ ⊄ A for A = φ
e)   φ ⊆ A      	T      	φ ⊆ A ∀A
f)   φ ⊆ ℘(A)         	T      	φ ⊆ A ∀A
g)   {θ} ⊆ ℘(φ)       	F       {θ} ⊄ ℘(φ) ℘(φ) = θ
h)   θ∩φ = φ           	T      	θ∩φ = φ
i)   ℘(φ) = {φ,θ}     	F      	℘(φ) = θ ≠ {φ,θ}
j)   φ ∈ A             	F      	φ ∉ φ for A = φ
k)   φ ∈ ℘(A)         	T      	φ ∈ ℘(A) ∀A
l)   θ ∈ ℘(A)        	F      	θ ∉ θ for A = φ
m)   θ∪φ = θ          	T      	θ∪φ = θ
n)   φ ⊆ ℘(F)-φ       	T      	φ ⊆ A ∀A
o)   θ ⊆ {{φ,θ,{θ}}}  	F      	φ ≠ {φ,θ,{θ}}
These are my answers and only a few from

"Bridge to Abstract Mathematics",1987
 by Ronald P. Morash.
  

Comments: ( You must be logged in to post comments.)
  Subject Author Date
Some Thoughtsre: attempt at the restCharlie2014-04-19 14:49:15
Some Thoughtsattempt at the restCharlie2014-04-19 11:01:51
Some Thoughtsright off the batCharlie2014-04-19 09:39:42
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