Subset

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, $A\subseteq B$ is represented as $\forall x\left(x\in A\Rightarrow x\in B\right).$ ^[1]

One can prove the statement $A\subseteq B$ by applying a proof technique known as the element argument^[2]:

Let sets A and B be given. To prove that $A\subseteq B,$
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 $(c\in A)\Rightarrow (c\in B)$ for an arbitrarily chosen element c. Universal generalisation then implies $\forall x\left(x\in A\Rightarrow x\in B\right),$ which is equivalent to $A\subseteq B,$ as stated above.

^ Rosen, Kenneth H. (2012). Discrete Mathematics and Its Applications (7th ed.). New York: McGraw-Hill. p. 119. ISBN 978-0-07-338309-5.
^ Epp, Susanna S. (2011). Discrete Mathematics with Applications (Fourth ed.). p. 337. ISBN 978-0-495-39132-6.

[1] Rosen, Kenneth H. (2012). Discrete Mathematics and Its Applications (7th ed.). New York: McGraw-Hill. p. 119. ISBN 978-0-07-338309-5.

[2] Epp, Susanna S. (2011). Discrete Mathematics with Applications (Fourth ed.). p. 337. ISBN 978-0-495-39132-6.

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