Set theory is now part of mathematics. We all know that a set is called any collection of elements clearly distinguishable from each other, which have one (or more) characteristics in common. Set theory studies the properties and relationships of sets; This field was promoted by Bolzano and Cantor, later perfected already in the 20th century by other mathematicians, such as Zermelo and Fraenkel.
It is important that every set is perfectly defined, that is, it can be established with precision whether given an object, it belongs or not to the set.
- In mathematics this is generally straightforward. For example, if the set of even numbers greater than 1 and less than 15 is considered, it is clear that this set will consist of the digits 2, 4, 6, 8, 10, 12 and 14 only.
- At common language, talking about a group can be much more imprecise, because if we want to form the group of the best singers, for example, opinions will be diverse and there will be no absolute consensus on who will be part of this group and who will not. Some special sets are empty sets (devoid of elements) or unitary sets (with only one element).
The objects that are part of a set are called members or elements, and sets are represented in written texts enclosed in braces: {}. Inside the brace, items are separated by commas. They can also be represented by Venn diagrams, which enclose the collections of elements that make up each set in a solid and closed line, generally in the shape of a circle. When there are several of these closed lines, each of them is assigned a capital letter (A, B, C, etc.) and the global set of these is represented by the letter U, which means universal set.
With sets you can perform operations; the main ones are union, intersection, difference, complement and Cartesian product. The union of two sets A and B is defined as the set A ∪ B and this contains each element that is in at least one of them. The general equation that represents it is:
- TO= {José, Jerónimo}, B= {María, Mabel, Marcela}; AUB= {José, Jerónimo, María, Mabel, Marcela}
- P= {pear, apple}, C= {lemon, orange}; F= {cherry, currant};PUCUF = {pear, apple, lemon, orange, cherry, currant}
- M={7, 9, 11}, N={4, 6, 8}; MUN={7, 9, 11, 4, 6, 8}
- R= {ball, skate, paddle}, G= {paddle, ball, skate}; RUG= {ball, paddle, skate}
- C= {daisy}, S= {carnation}; CUS = {daisy, carnation}
- C= {daisy}, S= {carnation}; T= {bottle}, CUSUT = {margarita, carnation, bottle}
- G= {green, blue, black}, H= {black}; GUH= {green, blue, black}
- TO={ 1, 3, 5, 7, 9 }; B={ 10, 11, 12 }; AUB={ 1, 3, 5, 7, 9, 10, 11, 12 }
- D= {Tuesday, Thursday}, AND= {Wednesday, Friday}; DUE = {Tuesday, Wednesday, Thursday, Friday}
- B= {mosquito, bee, hummingbird}; C= {cow, dog, horse}; BUC= {mosquito, bee, hummingbird, cow, dog, horse}
- TO={2, 4, 6, 8}, B={1, 2, 3, 4}; AUB={1, 2, 3, 4, 6, 8}
- P= {table, chair}, Q= {table, chair}; PUQ= {table, chair}
- TO= {bread}, B = {cheese}; AUB= {bread, cheese}
- TO={20, 30, 40}, B= {5, 15}; AUB ={5, 15, 20, 30, 40}
- M= {January, February, March, April}, N= {November, December}; MUN= {January, February, March, April, November, December}
- F={12, 22, 32, 42}, G= {a, e, i, o, u}; FUG= {12, 22, 32, 42, a, e, i, o, u}
- TO= {summer}, B= {winter}; AUB= {summer, winter}
- S= {sandal, slipper, flip flop}, R= {shirt}; SOUTH= {sandal, slipper, flip flop, shirt}
- H= {Monday, Tuesday}, R= {Monday, Tuesday}, D= {Monday, Tuesday}; HURUD= {Monday, Tuesday}
- P= {red, blue}, Q= {green, yellow}, PUQ= {red, blue, green, yellow}
