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Please solve this question along with explanation.

in Mathematical Logic by Junior (965 points)
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a) true  b) true c) false d)true e)true f)true g)false

few points

1) contains means set has element .

2) set A is said to be proper  subset of B , if there exist element of B that is not in set A.(A$\subset$ B)

NOW in option  a, b,c ,d asking about lhs element contain in rhs side set ...

in option c , lhs itself  is element  $\left \{ \Phi \right \}$ but rhs has set so its false

in option e ) proper subset of rhs will be $\left \{ \Phi \right \}, \left \{ \left \{ \Phi \right \} \right \}$ so its true

similar f ) is true

for g) we have to find proper subset of $\left \{ \left \{ \Phi \right \},\left \{ \Phi \right \} \right \}= \left \{ \left \{ \Phi \right \} \right \}$

so lhs will never be propersubset its itself set ...
by Boss (17.1k points)
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I think option G is false because it has two same elements in the set, which disobeys the distinct property of sets.
0
thats why i wrote in single element set , so this set is not proper set of itself
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(G) is not set according to Wikipedia definition of set is: 

In mathematics, a set is a collection of distinct objects.

 Since it is not a set, so operations on sets are also meaningless.

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