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Venn diagrams (or set diagrams) are illustrations used in the branch of mathematics known as set theory. Invented in 1881 by John Venn, they show all of the possible mathematical or logical relationships between sets (groups of things). They normally consist of overlapping circles. For instance, in a two-set Venn diagram, one circle may represent the group of all wooden objects, while another circle may represent the set of all tables. The overlapping area (intersection) would then represent the set of all wooden tables. Shapes other than circles can be employed (see below), and this is necessary for more than three sets. The following example involves two sets, A and B, represented here as coloured circles. The orange circle, set A, represents all 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 both can fly and have two legs — for example, parrots — are then in both sets, so they correspond to points in the area where the blue and orange circles overlap. That area contains all such and only such living creatures. Humans and penguins are bipedal, and so are then 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. The combined area of sets A and B is called the union of A and B, denoted by A ? B. The union in this case contains all things that either have two legs, or that fly, or both.
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