Can a finite set be equivalent to a proper subset of itself?

Write the proper subsets of {2,4,6}

Write the proper subsets of {1,2,3}

Write the proper subsets of {1,3,5}

Is inclusion of a subset in another, in the context of a universal set, an equivalence relation in the class of subsets of the sets ? Justify your answer.

Is inclusion of a subset in another, in the context of a universal set, an equivalence relation in the class of subsets of the sets ? Justify your answer.

A is a set containing n elements. A subset P of A is chosen. The set A is reconstructed by replacing the elements of P. A subset Q of A is again chosen, the number of ways of choosing so that (P cup Q) is a proper subset of A, is

Let A be a finite set. Then, each injective function from A into itself is not surjective.

Two finite sets have m and n elements. The total number of subsets of the first set is 56 more than the total number of subsets of the second set. Find the values of m and n.

Two finite sets have m and n elements. The total number of subsets of the first set is 48 more than the total number of subsets of the second set. The value of m - n is