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Empty set

In mathematics, the empty set is the unique set having no elements. Some axiomatic set theories ensure. Many possible properties of sets are vacuously true for the empty set. In some textbooks and popularizations, the empty set is referred to as the "null set"; however null set is a distinct notion within the context of measure theory. In that setting, it describes a set of measure zero; the empty set may be called the void set. Common notations for the empty set include "", " ∅ ", "∅". Common notations for the empty set include "", " ∅ ", "∅"; the latter two symbols were introduced by the Bourbaki group in 1939, inspired by the letter Ø in the Norwegian and Danish alphabets. In the past, "0" was used as a symbol for the empty set, but this is now considered to be an improper use of notation; the symbol ∅ is available at Unicode point U+2205. It can be coded in HTML, it can be coded in LaTeX as \varnothing. The symbol ∅ is coded in LaTeX as \emptyset. In standard axiomatic set theory, by the principle of extensionality, two sets are equal if they have the same elements.

Hence there is but one empty set, we speak of "the empty set" rather than "an empty set". For any set A: The empty set is a subset of A: ∀ A: ∅ ⊆ A The union of A with the empty set is A: ∀ A: A ∪ ∅ = A The intersection of A with the empty set is the empty set: ∀ A: A ∩ ∅ = ∅ The Cartesian product of A and the empty set is the empty set: ∀ A: A × ∅ = ∅ The empty set has the following properties: Its only subset is the empty set itself: ∀ A: A ⊆ ∅ ⇒ A = ∅ The power set of the empty set is the set containing only the empty set: 2 ∅ = Its number of elements is zero: | ∅ | = 0 The connection between the empty set and zero goes further, however: in the standard set-theoretic definition of natural numbers, sets are used to model the natural numbers. In this context, zero is modelled by the empty set. For any property: For every element of ∅ the property holds. Conversely, if for some property and some set V, the following two statements hold: For every element of V the property holds There is no element of V for which the property holdsthen V = ∅.

By the definition of subset, the empty set is a subset of any set A. That is, every element x of ∅ belongs to A. Indeed, if it were not true that every element of ∅ is in A there would be at least one element of ∅, not present in A. Since there are no elements of ∅ at all, there is no element of ∅, not in A. Any statement that begins "for every element of ∅ " is not making any substantive claim; this is paraphrased as "everything is true of the elements of the empty set." When speaking of the sum of the elements of a finite set, one is led to the convention that the sum of the elements of the empty set is zero. The reason for this is; the product of the elements of the empty set should be considered to be one, since one is the identity element for multiplication. A derangement is a permutation of a set without fixed points; the empty set can be considered a derangement of itself, because it has only one permutation, it is vacuously true that no element can be found that retains its original position.

Since the empty set has no members, when it is considered as a subset of any ordered set every member of that set will be an upper bound and lower bound for the empty set. For example, when considered as a subset of the real numbers, with its usual ordering, represented by the real number line, every real number is both an upper and lower bound for the empty set; when considered as a subset of the extended reals formed by adding two "numbers" or "points" to the real numbers, namely negative infinity, denoted − ∞, defined to be less than every other extended real number, positiv

Maharashtra Kesari

Maharashtra Kesari is an Indian-style wrestling championship, established in 1961 in the Indian state of Maharashtra. Cash prize was awarded to the winner since the start, but since 1982, initiated by wrestler Mamasaheb Mohol, the winner was permanently awarded with a silver mace weighing 1.5 kilograms. As of 2008 the winner received a silver mace and a cash prize of ₹51,000. In addition, Government of Maharashtra provided honorary sum amount to the title holders above age 50; until 2012, to be eligible for this amount, the title holder's annual income had to be less than₹40,000. In 2015, the cash prize was raised to ₹1 lakh; the winners are eligible to apply for government jobs under sports quota. Harshwardhan Sadgir is current Maharashtra Kesri. Bala Rafik became the winner of 2018 by defeating the winner of the 2017 edition Abhijit Katke. Narsing Yadav holds the record of winning the title three times consecutively from 2011 to 2013 and in 2016 Vijay chaudhary tied the record of Narsing Yadav.

Vijay won three times consecutively from 2014 to 2016, whereas Ganpat Khedkar, Chamba Mutnaal, Laxman Wadar, Dadu Chougule and Chandrahar Patil have won the title twice. After winning his second title in 2008, Chandrahar Patil competed in the 2009 championship whereas previous double title holders had not competed the third time. Yuvaraj Patil hold the record of being the youngest winner at the age of 17. Key ^ Hiraman Bankar is father of Vijay Bankar. Dadu Chaugule is father of vinod Chaughule. विजय चौधरी ठरला'महाराष्ट्र केसरी' |}

Ignalina District Municipality

Ignalina District Municipality is one of 60 municipalities in Lithuania. District structure: 2 cities – Dūkštas and Ignalina. Ignalina District Municipality consists of 12 smaller administration units - elderships. Population of largest Ignalina District Municipality elderships: Ignalina town – 5605 Didžiasalis – 1691 Vidiškės – 1278 Dūkštas – 1756 Kazitiškis – 1039 Naujasis Daugėliškis – 1491 Mielagėnai – 887 Ceikiniai – 533 Linkmenys – 970 Rimšė – 999 Tverečius – 590In total - 18414 inhabitants. Aukštaitija National Park