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These are two examples in which both the subset and the whole set are infinite, and the subset has the same cardinality (the concept that corresponds to size, that is, the number of elements, of a finite set) as the whole; such cases can run counter to one's initial intuition.
The red subset = {1,2,3,4} has two maximal elements, viz. 3 and 4, and one minimal element, viz. 1, which is also its least element. In mathematics , especially in order theory , a maximal element of a subset S {\displaystyle S} of some preordered set is an element of S {\displaystyle S} that is not smaller than any other element in S ...
These elements are also maximal and minimal elements, respectively, of the red subset. In mathematics, especially in order theory, the greatest element of a subset of a partially ordered set (poset) is an element of that is greater than every other element of .
For example the elements of the set = {,,} are the color red, the number 12, and the set B ... Thus the relation is a subset of U × P(U). The converse ...
For example, {1, 2} is a subset of {1, 2, 3}, and so is {2} but {1, 4} is not. As implied by this definition, a set is a subset of itself. For cases where this possibility is unsuitable or would make sense to be rejected, the term proper subset is defined. A is called a proper subset of B if and only if A is a subset of B, but A is not equal to B.
An antichain is a subset of a poset in which no two distinct elements are comparable. For example, the set of singletons {{}, {}, {}}. An element a is said to be strictly less than an element b, if a ≤ b and .
Infinite groups can also have finite generating sets. The additive group of integers has 1 as a generating set. The element 2 is not a generating set, as the odd numbers will be missing. The two-element subset {3, 5} is a generating set, since (−5) + 3 + 3 = 1 (in fact, any pair of coprime numbers is, as a consequence of Bézout's identity).
Counting the empty set as a subset, a set with elements has a total of subsets, and the theorem holds because > for all non-negative integers. Much more significant is Cantor's discovery of an argument that is applicable to any set, and shows that the theorem holds for infinite sets also.