Subsets
In mathematics, a set A is a subset of a set B 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
{\displaystyle A\subseteq B}
is represented as
∀
x
(
x
∈
A
⇒
x
∈
B
)
.
{\displaystyle \forall x\left(x\in A\Rightarrow x\in B\right).}
One can prove the statement
A
⊆
B
{\displaystyle A\subseteq B}
by applying a proof technique known as the element argument:Let sets A and B be given. To prove that
A
⊆
B
,
{\displaystyle A\subseteq 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
)
{\displaystyle (c\in A)\Rightarrow (c\in B)}
for an arbitrarily chosen element c. Universal generalisation then implies
∀
x
(
x
∈
A
⇒
x
∈
B
)
,
{\displaystyle \forall x\left(x\in A\Rightarrow x\in B\right),}
which is equivalent to
A
⊆
B
,
{\displaystyle A\subseteq B,}
as stated above.
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