Naive set theory
Subsets

Given two sets A and B, A is a subset of B if every element of A is also an element of B.In particular, each set B is a subset of itself; a subset of B that is not equal to B is called a proper subset.

If A is a subset of B, then one can also say that B is a superset of A, that A is contained in B, or that B contains A. In symbols, A ⊆ B means that A is a subset of B, and B ⊇ A means that B is a superset of A.Some authors use the symbols ⊂ and ⊃ for subsets, and others use these symbols only for proper subsets. For clarity, one can explicitly use the symbols ⊊ and ⊋ to indicate non-equality.

As an illustration, let R be the set of real numbers, let Z be the set of integers, let O be the set of odd integers, and let P be the set of current or former U.S. Presidents.Then O is a subset of Z, Z is a subset of R, and (hence) O is a subset of R, where in all cases subset may even be read as proper subset.Not all sets are comparable in this way. For example, it is not the case either that R is a subset of P nor that P is a subset of R.

It follows immediately from the definition of equality of sets above that, given two sets A and B, A = B if and only if A ⊆ B and B ⊆ A. In fact this is often given as the definition of equality. Usually when trying to prove that two sets are equal, one aims to show these two inclusions. The empty set is a subset of every set (the statement that all elements of the empty set are also members of any set A is vacuously true).

The set of all subsets of a given set A is called the power set of A and is denoted by 2 A {\displaystyle 2^{A}} or P (A) {\displaystyle P(A)}; the " P " is sometimes in a script font: ℘ (A) {\displaystyle \wp (A)}. If the set A has n elements, then P (A) {\displaystyle P(A)} will have 2 n {\displaystyle 2^{n}} elements.
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