Subsets
In mathematics, a set A is a subset of a set B if and only if all elements of A are also elements of B; B is then a superset of A. It is possible for A and B to be equal; if they are unequal, then A is a proper subset of B. The relationship of one set being a subset of another is called inclusion (or sometimes containment). A is a subset of B may also be expressed as B includes (or contains) A or A is included (or contained) in B. A k-subset is a subset with k elements.
When quantified, A⊆B is represented as ∀x(x∈A⇒x∈B).
One can prove the statement A⊆B by applying a proof technique known as the element argument:Let sets A and B be given. To prove that A⊆B,
suppose that a is a particular but arbitrarily chosen element of A
show that a is an element of B.
The validity of this technique can be seen as a consequence of universal generalization: the technique shows (c∈A)⇒(c∈B) for an arbitrarily chosen element c. Universal generalisation then implies ∀x(x∈A⇒x∈B), which is equivalent to A⊆B, as stated above.
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