Discrete Mathematics

Question

Is EMPTY SET ( $\phi$ ) is Proper subset or Improper subset of Non-empty set ??

Comments

$\phi$ is a subset of any set. It is also always a proper subset of any set except itself.

 $\phi\subseteq\phi$

$but$  $\phi$ ⊄$\phi$ So,this is improper subset of itself.

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For non empty set

Improper subset is Only set itself rest all are proper subsets

For  empty set

Only one subset which is empty set and it is improper subset of that set

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if  $\phi$ is proper subset

Then in question https://gateoverflow.in/2428/gate2007-59

How answer is C which is "Courses for which only proper subsets of female enrolled

So it can be also in output "Courses for which no girl enrolled"

But in question it is given that 

 every course is taken by at least one male and at least one female student

 

Answer 1

Proper Subset
empty set is a subset of any set and the non empty set contains atleast one element that is not in empty set
so.. it s  proper subset