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2. Element (mathematics) - Wikipedia

   en.wikipedia.org/wiki/Element_(mathematics)

   In mathematics, an element (or member) ... while a finite set is a set with a finite number of elements. The above examples are examples of finite sets.

3. Set (mathematics) - Wikipedia

   en.wikipedia.org/wiki/Set_(mathematics)

   A set of polygons in an Euler diagram This set equals the one depicted above since both have the very same elements.. In mathematics, a set is a collection of different [1] things; [2] [3] [4] these things are called elements or members of the set and are typically mathematical objects of any kind: numbers, symbols, points in space, lines, other geometrical shapes, variables, or even other ...

4. Greatest element and least element - Wikipedia

   en.wikipedia.org/wiki/Greatest_element_and_least...

   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 . The term least element is defined dually , that is, it is an element of S {\displaystyle S} that is smaller than every other element of S . {\displaystyle S.}

5. Maximal and minimal elements - Wikipedia

   en.wikipedia.org/wiki/Maximal_and_minimal_elements

   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 of some preordered set is an element of that is not smaller than any other element in .

6. Partition of a set - Wikipedia

   en.wikipedia.org/wiki/Partition_of_a_set

   Each set of elements has a least upper bound (their "join") and a greatest lower bound (their "meet"), so that it forms a lattice, and more specifically (for partitions of a finite set) it is a geometric and supersolvable lattice. [6] [7] The partition lattice of a 4-element set has 15 elements and is depicted in the Hasse diagram on the left.

7. Partially ordered set - Wikipedia

   en.wikipedia.org/wiki/Partially_ordered_set

   As another example, consider the positive integers, ordered by divisibility: 1 is a least element, as it divides all other elements; on the other hand this poset does not have a greatest element. This partially ordered set does not even have any maximal elements, since any g divides for instance 2 g , which is distinct from it, so g is not maximal.

8. Field (mathematics) - Wikipedia

   en.wikipedia.org/wiki/Field_(mathematics)

   The above introductory example F 4 is a field with four elements. Its subfield F 2 is the smallest field, because by definition a field has at least two distinct elements, 0 and 1 . In modular arithmetic modulo 12, 9 + 4 = 1 since 9 + 4 = 13 in Z , which divided by 12 leaves remainder 1 .

9. Order (group theory) - Wikipedia

   en.wikipedia.org/wiki/Order_(group_theory)

   For example, in the symmetric group shown above, where ord(S 3) = 6, the possible orders of the elements are 1, 2, 3 or 6. The following partial converse is true for finite groups : if d divides the order of a group G and d is a prime number , then there exists an element of order d in G (this is sometimes called Cauchy's theorem ).