Subset

Article summary

showing A is a subset of B (denoted A \subseteq B ) and, conversely, B is a superset of A (denoted B \supseteq A ) 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 \subseteq B is represented as \forall x \left(x \in A \Rightarrow x \in B\right). One can prove the statement A \subseteq B by applying a proof technique known as the element argument : 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.

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1. Summarise the article in your own words. Which single claim would you verify first, and why?
2. Which claim in this article is most likely to be revised as the sources update? How would you check its current status?
3. If two sources disagreed about one of these facts, what would you need in order to trust one over the other?

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