Trivial Subset-

The trivial subsets of a set are those subsets which can be found without knowing the contents of the set. 

The empty set has one trivial subset: the empty set. 

Every nonempty set S has two distinct trivial subsets: S and the empty set. 

Non trivial Subset-

A subset of a set which is neither the empty set nor the entirety of the set. 

To demonstrate nontriviality of a subset it is necessary and sufficient to demonstrate that the subset contains at least one element and that there is at least one element of the superset which is not contained in the subset.

For example-

Reptiles are a trivial subset of the set reptiles as would be a crocoduck. 

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