File Name: venn diagram and set operations .zip
It is natural for us to classify items into groups, or sets, and consider how those sets overlap with each other.
set operations pdf
A Venn diagram is a widely-used diagram style that shows the logical relation between sets , popularised by John Venn in the s.
The diagrams are used to teach elementary set theory , and to illustrate simple set relationships in probability , logic , statistics , linguistics and computer science. A Venn diagram uses closed curves drawn on a plane to represent sets. Very often, these curves are circles or ellipses. Similar ideas had been proposed before Venn. A Venn diagram may also be called a primary diagram , set diagram or logic diagram.
It is a diagram that shows all possible logical relations between a finite collection of different sets. These diagrams depict elements as points in the plane, and sets as regions inside closed curves. A Venn diagram consists of multiple overlapping closed curves, usually circles, each representing a set. The points inside a curve labelled S represent elements of the set S , while points outside the boundary represent elements not in the set S. In Venn diagrams, the curves are overlapped in every possible way, showing all possible relations between the sets.
They are thus a special case of Euler diagrams , which do not necessarily show all relations. Venn diagrams were conceived around by John Venn. They are used to teach elementary set theory , as well as illustrate simple set relationships in probability , logic , statistics , linguistics , and computer science.
A Venn diagram in which the area of each shape is proportional to the number of elements it contains is called an area-proportional or scaled Venn diagram. Venn diagrams are used heavily in the logic of class branch of reasoning. This example involves two sets , A and B, represented here as coloured circles. The orange circle, set A, represents all types of living creatures that are two-legged. The blue circle, set B, represents the living creatures that can fly.
Each separate type of creature can be imagined as a point somewhere in the diagram. Living creatures that can fly and have two legs—for example, parrots—are then in both sets, so they correspond to points in the region where the blue and orange circles overlap.
This overlapping region would only contain those elements in this example, creatures that are members of both set A two-legged creatures and set B flying creatures. Humans and penguins are bipedal, and so are in the orange circle, but since they cannot fly, they appear in the left part of the orange circle, where it does not overlap with the blue circle.
Mosquitoes have six legs, and fly, so the point for mosquitoes is in the part of the blue circle that does not overlap with the orange one.
Creatures that are not two-legged and cannot fly for example, whales and spiders would all be represented by points outside both circles. Venn diagrams were introduced in by John Venn in a paper entitled "On the Diagrammatic and Mechanical Representation of Propositions and Reasonings" in the Philosophical Magazine and Journal of Science , about the different ways to represent propositions by diagrams.
They are rightly associated with Venn, however, because he comprehensively surveyed and formalized their usage, and was the first to generalize them". Venn himself did not use the term "Venn diagram" and referred to his invention as " Eulerian Circles ". Of these schemes one only, viz.
Venn diagrams are very similar to Euler diagrams , which were invented by Leonhard Euler in the 18th century. Baron has noted that Leibniz — produced similar diagrams before Euler in the 17th century, but much of it was unpublished.
In the 20th century, Venn diagrams were further developed. David Wilson Henderson showed, in , that the existence of an n -Venn diagram with n -fold rotational symmetry implied that n was a prime number. These combined results show that rotationally symmetric Venn diagrams exist, if and only if n is a prime number. Venn diagrams and Euler diagrams were incorporated as part of instruction in set theory , as part of the new math movement in the s. Since then, they have also been adopted in the curriculum of other fields such as reading.
In , sexologist Dr. Lindsey Doe began a trend of using the word " cunt " to refer to the intersection of A and B in a Venn diagram. This was an attempt to add this usage of the word to a dictionary by A Venn diagram is constructed with a collection of simple closed curves drawn in a plane. According to Lewis,  the "principle of these diagrams is that classes [or sets ] be represented by regions in such relation to one another that all the possible logical relations of these classes can be indicated in the same diagram.
That is, the diagram initially leaves room for any possible relation of the classes, and the actual or given relation, can then be specified by indicating that some particular region is null or is not-null". Venn diagrams normally comprise overlapping circles. The interior of the circle symbolically represents the elements of the set, while the exterior represents elements that are not members of the set.
For instance, in a two-set Venn diagram, one circle may represent the group of all wooden objects, while the other circle may represent the set of all tables.
The overlapping region, or intersection , would then represent the set of all wooden tables. Shapes other than circles can be employed as shown below by Venn's own higher set diagrams. Venn diagrams do not generally contain information on the relative or absolute sizes cardinality of sets. That is, they are schematic diagrams generally not drawn to scale.
Venn diagrams are similar to Euler diagrams. However, a Venn diagram for n component sets must contain all 2 n hypothetically possible zones, that correspond to some combination of inclusion or exclusion in each of the component sets. In Venn diagrams, a shaded zone may represent an empty zone, whereas in an Euler diagram, the corresponding zone is missing from the diagram.
For example, if one set represents dairy products and another cheeses , the Venn diagram contains a zone for cheeses that are not dairy products. Assuming that in the context cheese means some type of dairy product, the Euler diagram has the cheese zone entirely contained within the dairy-product zone—there is no zone for non-existent non-dairy cheese.
This means that as the number of contours increases, Euler diagrams are typically less visually complex than the equivalent Venn diagram, particularly if the number of non-empty intersections is small. The difference between Euler and Venn diagrams can be seen in the following example.
Take the three sets:. Venn diagrams typically represent two or three sets, but there are forms that allow for higher numbers. Shown below, four intersecting spheres form the highest order Venn diagram that has the symmetry of a simplex and can be visually represented.
The 16 intersections correspond to the vertices of a tesseract or the cells of a cell , respectively. For higher numbers of sets, some loss of symmetry in the diagrams is unavoidable. Venn was keen to find "symmetrical figures He also gave a construction for Venn diagrams for any number of sets, where each successive curve that delimits a set interleaves with previous curves, starting with the three-circle diagram.
Non-example: This Euler diagram is not a Venn diagram for four sets as it has only 13 regions excluding the outside ; there is no region where only the yellow and blue, or only the red and green circles meet.
Anthony William Fairbank Edwards constructed a series of Venn diagrams for higher numbers of sets by segmenting the surface of a sphere, which became known as Edwards—Venn diagrams.
A fourth set can be added to the representation, by taking a curve similar to the seam on a tennis ball, which winds up and down around the equator, and so on. The resulting sets can then be projected back to a plane, to give cogwheel diagrams with increasing numbers of teeth—as shown here.
These diagrams were devised while designing a stained-glass window in memory of Venn. They are also two-dimensional representations of hypercubes. Henry John Stephen Smith devised similar n -set diagrams using sine curves  with the series of equations. Charles Lutwidge Dodgson a. Lewis Carroll devised a five-set diagram known as Carroll's square.
Joaquin and Boyles, on the other hand, proposed supplemental rules for the standard Venn diagram, in order to account for certain problem cases. For instance, regarding the issue of representing singular statements, they suggest to consider the Venn diagram circle as a representation of a set of things, and use first-order logic and set theory to treat categorical statements as statements about sets. Additionally, they propose to treat singular statements as statements about set membership.
So, for example, to represent the statement "a is F" in this retooled Venn diagram, a small letter "a" may be placed inside the circle that represents the set F. Another way of representing sets is with John F.
Randolph's R-diagrams. From Wikipedia, the free encyclopedia. Diagram that shows all possible logical relations between a collection of sets. Math Vault. Retrieved Archived PDF from the original on Proceedings of the Cambridge Philosophical Society.
MAA Online. The Electronic Journal of Combinatorics. A Survey of Symbolic Logic. Berkeley: University of California Press. Symbolic logic. McMaster University. Archived from the original PDF on Has a detailed history of the evolution of logic diagrams including but not limited to the Venn diagram. In Couturat, Louis ed. Opuscules et fragmentes inedits de Leibniz in Latin.
May The Mathematical Gazette. American Mathematical Monthly. Notices of the AMS. Archived from the original on
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Just because it worked for these, doesn't mean you can assume everything is the same. The purpose of this module is to introduce language for talking about sets, and some Set operations can be used to combine sets. Turret lathes and special purpose lathes are usually used in production or job shops for Let. Set Operations. We Sets.
Sets are treated as mathematical objects. Similarly to numbers, we can perform certain mathematical operations on sets. Below we consider the principal operations involving the intersection , union , difference , symmetric difference , and the complement of sets. To visualize set operations, we will use Venn diagrams. In a Venn diagram, a rectangle shows the universal set, and all other sets are usually represented by circles within the rectangle. The shaded region represents the result of the operation.
5.1: Sets and Operations on Sets
A Venn diagram is a widely-used diagram style that shows the logical relation between sets , popularised by John Venn in the s. The diagrams are used to teach elementary set theory , and to illustrate simple set relationships in probability , logic , statistics , linguistics and computer science. A Venn diagram uses closed curves drawn on a plane to represent sets. Very often, these curves are circles or ellipses. Similar ideas had been proposed before Venn.
Let us discuss the important operations here: The important operations on sets are. Set operations in LINQ refer to query operations that produce a result set that is based on the presence or absence of equivalent elements within the same or separate collections or sets. For any one of the set operations, we can expand to set builder notation, and then use the logical equivalences to manipulate the conditions. Since we're doing the same manipulations, we ended up with the same tables. Create a Venn diagram to show the relationship among the sets.
Each friend is an "element" or "member" of the set. It is normal to use lowercase letters for them.
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