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Consider the same example above. If the element 4 4 is removed from the set B B, solve for (i), (ii), (iii) as well. (i) Since there is no common elements in sets A A and B B, then A \cap B = \phi A∩B = ϕ or A \cap B = \ { \} A∩B = {}. (ii) Because the element 4 4 is no longer repeated, then A \cup B A ∪B remains the same.
- Rough Definition
- Relation of Belonging
- Subsets
- Operations
- Infinite Sets
- Cardinality
- Problems
- External Links
A set is a collection of objects. The objects can be anything: numbers, letters, libraries that have at least 20 male staff, or absolutely nothing. Order does not matter. What does matter is what is in the set. There might be a finite number of objects in the set, in which case it is called a finite set. Otherwise we call it an infinite set. The ob...
The most important property of sets is that, for every object and a set , we can say whether belongs to (written as ), or not (written as ). Two sets and are equal if they include the same objects, i.e., if for every object , we have if and only if .
We say that a set is a subset of a set if every object that belongs to also belongs to . This is denoted or . For example, the sets and are subsets of the set , but the set is not. Thus we can say that two sets are equal if and only if each is a subset of the other. A special kind of subset is the empty set.
Union and Intersection
The union of two or more sets is the set of all objects that belong to one or more of the sets. The union of A and B is denoted . For example, the union of and is . Unions can also be represented just as sums and products can be. would be the union of all sets that satisfy the statement. So, for example, would be the set of all natural numbers . The intersection of two or more sets is the set of all objects that belong to all of the sets. The intersection of A and B is denoted . For example,...
Cartesian Product
The Cartesian Product of two sets and is defined as the set of Ordered Pairs such that and
An infinite set can be defined as a set that has the same cardinality as one of its proper subsets. Alternatively, infinite sets are those which cannot be put into correspondence with any set of the form .
The cardinality of a set , denoted , is (informally) the size of the set. For a finite set, the cardinality is simply the number of elements. The empty set has cardinality 0. iff there is a bijective function meaning that there is a function that maps all elements of to all the elements of with one-to-one correspondence. iff there exists an injecti...
Introductory
1. The regular 5-point star is drawn and in each vertex, there is a number. Each and are chosen such that all 5 of them came from set . Each letter is a different number (so one possible way is ). Let be the sum of the numbers on and , and so forth. If and form an arithmetic sequence (not necessarily in increasing order), find the value of . (Source)
Intermediate
1. Let set be a 90-element subset of and let be the sum of the elements of Find the number of possible values of (Source)
Olympiad
1. Let be a fixed positive integer, and let be an infinite family of sets, each of size , no two of which are disjoint. Prove that there exists a set of size that meets each set in . (Source)
May 27, 2024 · Set Theory is a branch of logical mathematics that studies the collection of objects and operations based on it. A set is simply a collection of objects or a group of objects. For example, a group of players in a football team is a set and the players in the team are its objects. The words collection, aggregate, and class are synonymous with set.
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Theorem 1.1.1 1.1. 1. Two sets A A and B B are equal if and only if A ⊂ B A ⊂ B and B ⊂ A B ⊂ A. If A ⊂ B A ⊂ B and A A does not equal B B, we say that A A is a proper subset of B B, and write A ⊊ B A ⊊ B. The set θ = {x: x ≠ x} θ = {x: x ≠ x} is called the empty set. This set clearly has no elements.
Sets may be thought of as a mathematical way to represent collections or groups of objects. The concept of sets is an essential foundation for various other topics in mathematics. This series of lessons cover the essential concepts of math set theory - the basic ways of describing sets, use of set notation, finite sets, infinite sets, empty ...
Oct 9, 2015 · Set theory has its own notations and symbols that can seem unusual for many. In this tutorial, we look at some solved examples to understand how set theory works and the kind of problems it can be used to solve. Definition. A set is a collection of objects. It is usually represented in flower braces. For example:
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A set is a collection of objects. The objects in a set are called its elements or members. The elements in a set can be any types of objects, including sets! The members of a set do not even have to be of the same type. For example, although it may not have any meaningful application, a set can consist of numbers and names.
