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A Venn diagram is a widely used diagram style that shows the logical relation between sets, popularized by John Venn (1834–1923) in the 1880s. The diagrams are used to teach elementary set theory, and to illustrate simple set relationships in probability, logic, statistics, linguistics and computer science.
The same method can also be illustrated with a Venn diagram as follows, with the prime factorization of each of the two numbers demonstrated in each circle and all factors they share in common in the intersection. The lcm then can be found by multiplying all of the prime numbers in the diagram. Here is an example: 48 = 2 × 2 × 2 × 2 × 3,
Venn diagram showing the union of sets A and B as everything not in white. In combinatorics, the inclusion–exclusion principle is a counting technique which generalizes the familiar method of obtaining the number of elements in the union of two finite sets; symbolically expressed as
A Karnaugh map (KM or K-map) is a diagram that can be used to simplify a Boolean algebra expression. Maurice Karnaugh introduced it in 1953 [ 1 ] [ 2 ] as a refinement of Edward W. Veitch 's 1952 Veitch chart , [ 3 ] [ 4 ] which itself was a rediscovery of Allan Marquand 's 1881 logical diagram [ 5 ] [ 6 ] (aka.
For MSO formulas that have free variables, when the input data is a tree or has bounded treewidth, there are efficient enumeration algorithms to produce the set of all solutions, [6] ensuring that the input data is preprocessed in linear time and that each solution is then produced in a delay linear in the size of each solution, i.e., constant ...
The algebra of sets is the set-theoretic analogue of the algebra of numbers. Just as arithmetic addition and multiplication are associative and commutative, so are set union and intersection; just as the arithmetic relation "less than or equal" is reflexive, antisymmetric and transitive, so is the set relation of "subset".
The first diagram in this article shows three circles (one for each parity bit) and encloses data bits that each parity bit covers. The second diagram (shown to the right) is identical but, instead, the bit positions are marked. For the remainder of this section, the following 4 bits (shown as a column vector) will be used as a running example:
By manipulating or inspecting a diagram, much tedious calculation may be eliminated. Graphical minimization methods for two-level logic include: Euler diagram (aka Eulerian circle) (1768) by Leonhard P. Euler (1707–1783) Venn diagram (1880) by John Venn (1834–1923) Karnaugh map (1953) by Maurice Karnaugh