yes proper subset is different from a regular subset. The empty set is a proper subset of every set except for the empty set
A proper subset of a set A is a subset of A that cannot be equal to A. In other words, if B is a proper subset of A, then all elements of B are in A but A contains at least one element that is not in B.
Subset: If A and B are sets and every element of A is also an element of B, then:
A is a subset of B, denoted by A ⊆ B.
or equivalently, B is a superset of A, denoted by B ⊇ A.
Any set is considered to be a subset of itself. No set is a proper subset of itself. The empty set is a subset of every set. The empty set is a proper subset of every set except for the empty set.
For example, A = {1, 2, 3, 4, 5} and B = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10}.
A = {1, 2, 3} and B = {1, 2, 3}
Here, A is a subset of B, or we can say that B is the superset of A.
Proper Subset: If A is a subset of B, but A is not equal to B (that is, there exists at least one element of B which is not an element of A), then
A is also a proper (or strict) subset of B; this is written as A ⊊ B.
For example A = {1, 2, 3} and B = {1, 2, 3, 4}.
Clearly, A is not equal to B and element {4} belongs to set B but is absent in set A, so we have one element in set B which is not an element of set A. Thus, A can be called a proper subset of B.
Hence, a proper subset of a set A is a subset of A that is not equal to A. In other words, if B is a proper subset of A, then all elements of B are in A but A contains at least one element that is not in B, thus the proper subset may or may not be the same.
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