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When is empty, the condition given above is an example of a vacuous truth. So the intersection of the empty family should be the universal set (the identity element for the operation of intersection), [4] but in standard set theory, the universal set does not exist.
A practical unit to be used by electricians and engineers, the ampere, was then defined as equal to one tenth of the electromagnetic unit of current. In another system, the "rationalized metre–kilogram–second (rmks) system" (or alternatively the "metre–kilogram–second–ampere (mksa) system"), k m is written as μ 0 /2 π , where μ 0 ...
A partition of a set X is a set of non-empty subsets of X such that every element x in X is in exactly one of these subsets [2] (i.e., the subsets are nonempty mutually disjoint sets). Equivalently, a family of sets P is a partition of X if and only if all of the following conditions hold: [ 3 ]
Let be a set and a nonempty family of subsets of ; that is, is a nonempty subset of the power set of . Then is said to have the finite intersection property if every nonempty finite subfamily has nonempty intersection; it is said to have the strong finite intersection property if that intersection is always infinite.
An optimal clique cover of the line graph () may be formed with one clique for each triangle in that has two or three degree-2 vertices, and one clique for each vertex that has degree at least two and is not a degree-two vertex of one of these triangles. The intersection number is the number of cliques of these two types.
Here, q 1 and q 2 are the charges, r is the distance between their centres, and the value of the constant fraction / is approximately 9 × 10 9 N⋅m 2 ⋅C −2. Likewise, ε 0 appears in Maxwell's equations , which describe the properties of electric and magnetic fields and electromagnetic radiation , and relate them to their sources.
As a result, the empty set is the unique initial object of the category of sets and functions. The empty set can be turned into a topological space, called the empty space, in just one way: by defining the empty set to be open. This empty topological space is the unique initial object in the category of topological spaces with continuous maps.
In mathematics, an empty product, or nullary product or vacuous product, is the result of multiplying no factors. It is by convention equal to the multiplicative identity (assuming there is an identity for the multiplication operation in question), just as the empty sum—the result of adding no numbers—is by convention zero, or the additive identity.