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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.
- Set Theory Definition
- History of Set Theory
- Examples of Sets
- Important Terms Related to Set Theory
- Types of Sets
- Set Theory Symbols
- Set Theory Formulas
- De Morgan’s Laws
- Visual Representation of Sets Using Venn Diagram
- Conclusion – Set Theory
Sets are defined as ”a well-defined collection of objects”. Let’s say we have a set of Natural Numbersthen it will have all the natural numbers as its members and the collection of the numbers is well defined that they are natural numbers. Note: A set is always denoted by a capital letter. A set of Natural Numbers is given by: The above example is ...
The concept of Set Theory was propounded in the year 1874 by Georg Cantorin his paper name ‘On a Property of Collection of All Real Algebraic Numbers‘. His concept of Set Theory was later used by other mathematicians in giving various other theories such as Klein’s Encyclopedia and Russell Paradox. Sets Theory is a foundation for a better understan...
Some common examples of sets are mentioned below: 1. Set of Natural Numbers: N = {1, 2, 3, 4….} 2. Set of Even Numbers: E = {2, 4, 6, 8…} 3. Set of Prime Numbers: P = {2, 3, 5, 7,….} 4. Set of Integers: Z = {…, -4, -3, -2, -1, 0, 1, 2,….}
Some of the important terms related to sets are mentioned below. These terms will be used several times in this article, and knowing these terms will help you learn set theory.
There are different types of sets categorized on various parameters. Some types of sets are mentioned below:
Various symbols are used in Sets Theory. The notations and their explanation are tabulated below: Read More: Set Theory Symbols.
The set theory formulasare given for two sets – overlapping and disjoint sets. Let’s learn them separately
De Morgan’s Lawis applicable in relating the union and intersection of two sets via their complements. There are two laws under De Morgan’s Law. Let’s learn them briefly
Venn Diagramis a technique for representing the relation between two sets with the help of circles, generally intersecting. For Example: Two circles intersecting with each other with the common area merged into them represent the union of sets, and two intersecting circles with a common area highlighted represent the intersection of sets while two ...
We have covered all the concepts required to learn the set theory. We have covered the history, definition, examples, symbols, operations, and formulas of set theory. Set theory is an important topic and many questions come from set theory in many competitive Exams. Students should focus on set theory and practice with some questions provided in th...
- 10 min
Example 2.6.6. If G is an open subset of R then every point of G is a limit point of G. Solution. In fact, more is true. If G is open and a ∈ G, then a is a limit point of G∖{a}. Indeed, let δ> 0 be such that B(a; δ) ⊂ G. Then (G∖{a}) ∩ B(a; δ) = (a − δ, a) ∪ (a, a + δ) and, thus B(a; δ) contains an infinite number of points ...
- Sets Definition. In mathematics, a set is defined as a well-defined collection of objects. Sets are named and represented using capital letters. In the set theory, the elements that a set comprises can be any kind of thing: people, letters of the alphabet, numbers, shapes, variables, etc.
- Representation of Sets in Set Theory. There are different set notations used for the representation of sets in set theory. They differ in the way in which the elements are listed.
- Sets Symbols. Set symbols are used to define the elements of a given set. The following table shows the set theory symbols and their meaning. Symbols. Meaning.
- Types of Sets. There are different types of sets in set theory. Some of these are singleton, finite, infinite, empty, etc. Singleton Sets. A set that has only one element is called a singleton set or also called a unit set.
In mathematics, a set is defined as a collection of distinct, well-defined objects forming a group. There can be any number of items, be it a collection of whole numbers, months of a year, types of birds, and so on. Each item in the set is known as an element of the set. We use curly brackets while writing a set.
- For disjoint sets A and B; $n(A U B) = n(A) + n(B)$ $A \cap B = \varnothing$ $n(A − B) = n(A)$
- No. There is one element inside the brackets. So, it is a singleton set, not a null set.
- Cartesian Product of Set A and B is defined as an ordered pair $(x, y)$ where $x \in A$ and $y \in B$.
- In real life, sets are used in bookshelves while arranging the books according to alphabetic order or genre; closets where dresses, tops, jeans are...
- Yes, the union of two sets includes the intersection of the sets. Union of set A and B includes the elements either in set A or in set B or in both...
Two examples: we could consider the set of all actors who have played The Doctor on Doctor Who, or the set of natural numbers between 1 and 10 inclusive. In the first case, Tom Baker is an element (or member) of the set, while Idris Elba, among many others, is not an element of the set. Also, the two examples are of different sets.
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Jan 24, 2021 · Suppose set A = {1,3,5} and set B = {5,5,3,3,1,1}. Sets A and B are equal because they contain the exact same elements, namely 1, 3, and 5. Remember, order and multiplicity of elements don’t matter. Let’s look at another example. Determine if the sets A and B are equal.
