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Two disjoint sets. In set theory in mathematics and formal logic, two sets are said to be disjoint sets if they have no element in common. Equivalently, two disjoint sets are sets whose intersection is the empty set. [1] For example, {1, 2, 3} and {4, 5, 6} are disjoint sets, while {1, 2, 3} and {3, 4, 5} are not disjoint. A collection of two ...
For symmetric difference, the sets ( ) and () = ( ) are always disjoint. So these two sets are equal if and only if they are both equal to ∅ . {\displaystyle \varnothing .} Moreover, L ∖ ( M R ) = ∅ {\displaystyle L\,\setminus \,(M\,\triangle \,R)=\varnothing } if and only if L ∩ M ∩ R = ∅ and L ⊆ M ∪ R . {\displaystyle L\cap M ...
In combinatorics, a laminar set family is a set family in which each pair of sets are either disjoint or related by containment. [1] [2] Formally, a set family {S 1, S 2, ...} is called laminar if for every i, j, the intersection of S i and S j is either empty, or equals S i, or equals S j. Let E be a ground-set of elements.
The cube is the only one of these whose Dehn invariant is zero. The Dehn invariants of each of the other four Platonic solids are unequal and nonzero. The Dehn invariant of the octahedron is − 2 {\displaystyle -2} times the Dehn invariant of a tetrahedron of the same edge length.
Set packing is a classical NP-complete problem in computational complexity theory and combinatorics, and was one of Karp's 21 NP-complete problems. Suppose one has a finite set S and a list of subsets of S. Then, the set packing problem asks if some k subsets in the list are pairwise disjoint (in other words, no two of them share an element).
A regular digon has both angles equal and both sides equal and is represented by Schläfli symbol {2}. It may be constructed on a sphere as a pair of 180 degree arcs connecting antipodal points, when it forms a lune. The digon is the simplest abstract polytope of rank 2. A truncated digon, t{2} is a square, {4}. An alternated digon, h{2} is a ...
The Bolyai–Gerwien theorem states that any polygon may be dissected into any other polygon of the same area, using interior-disjoint polygonal pieces. It is not true, however, that any polyhedron has a dissection into any other polyhedron of the same volume using polyhedral pieces (see Dehn invariant ).
The sets and are separated by closed neighbourhoods if there is a closed neighbourhood of and a closed neighbourhood of such that and are disjoint. Our examples, [ 0 , 1 ) {\displaystyle [0,1)} and ( 1 , 2 ] , {\displaystyle (1,2],} are not separated by closed neighbourhoods.