Equivalence classIn mathematics, when the elements of some set have a notion of equivalence (formalized as an equivalence relation), then one may naturally split the set into equivalence classes. These equivalence classes are constructed so that elements and belong to the same equivalence class if, and only if, they are equivalent. Formally, given a set and an equivalence relation on the of an element in denoted by is the set of elements which are equivalent to It may be proven, from the defining properties of equivalence relations, that the equivalence classes form a partition of This partition—the set of equivalence classes—is sometimes called the quotient set or the quotient space of by and is denoted by .
Absorbing elementIn mathematics, an absorbing element (or annihilating element) is a special type of element of a set with respect to a binary operation on that set. The result of combining an absorbing element with any element of the set is the absorbing element itself. In semigroup theory, the absorbing element is called a zero element because there is no risk of confusion with other notions of zero, with the notable exception: under additive notation zero may, quite naturally, denote the neutral element of a monoid.
Identity elementIn mathematics, an identity element or neutral element of a binary operation is an element that leaves unchanged every element when the operation is applied. For example, 0 is an identity element of the addition of real numbers. This concept is used in algebraic structures such as groups and rings. The term identity element is often shortened to identity (as in the case of additive identity and multiplicative identity) when there is no possibility of confusion, but the identity implicitly depends on the binary operation it is associated with.
Multiplicative group of integers modulo nIn modular arithmetic, the integers coprime (relatively prime) to n from the set of n non-negative integers form a group under multiplication modulo n, called the multiplicative group of integers modulo n. Equivalently, the elements of this group can be thought of as the congruence classes, also known as residues modulo n, that are coprime to n. Hence another name is the group of primitive residue classes modulo n. In the theory of rings, a branch of abstract algebra, it is described as the group of units of the ring of integers modulo n.
Set (mathematics)A set is the mathematical model for a collection of different things; a set contains elements or members, which can be mathematical objects of any kind: numbers, symbols, points in space, lines, other geometrical shapes, variables, or even other sets. The set with no element is the empty set; a set with a single element is a singleton. A set may have a finite number of elements or be an infinite set. Two sets are equal if they have precisely the same elements. Sets are ubiquitous in modern mathematics.
Note-takingNote-taking (sometimes written as notetaking or note taking) is the practice of recording information from different sources and platforms. By taking notes, the writer records the essence of the information, freeing their mind from having to recall everything. Notes are commonly drawn from a transient source, such as an oral discussion at a meeting, or a lecture (notes of a meeting are usually called minutes), in which case the notes may be the only record of the event.
AmitriptylineAmitriptyline, sold under the brand name Elavil among others, is a tricyclic antidepressant primarily used to treat major depressive disorder, a variety of pain syndromes such as neuropathic pain, fibromyalgia, migraine and tension headaches. Due to the frequency and prominence of side effects, amitriptyline is generally considered a second-line therapy for these indications. The most common side effects are dry mouth, drowsiness, dizziness, constipation, and weight gain. Of note is sexual dysfunction, observed primarily in males.
Haltlose personality disorderHaltlose personality disorder was a type of personality disorder diagnosis largely used in German-, Russian- and French-speaking countries. The German word haltlose refers to being "unstable" (literally: "without footing"), and in English-speaking countries the diagnosis was sometimes referred to as "the unstable psychopath" (although it was little known even among experts in psychiatry). In the early twentieth century, haltlose personality disorder was described by Emil Kraepelin and Gustav Aschaffenburg.
Order theoryOrder theory is a branch of mathematics that investigates the intuitive notion of order using binary relations. It provides a formal framework for describing statements such as "this is less than that" or "this precedes that". This article introduces the field and provides basic definitions. A list of order-theoretic terms can be found in the order theory glossary. Orders are everywhere in mathematics and related fields like computer science. The first order often discussed in primary school is the standard order on the natural numbers e.
Transitive relationIn mathematics, a relation R on a set X is transitive if, for all elements a, b, c in X, whenever R relates a to b and b to c, then R also relates a to c. Each partial order as well as each equivalence relation needs to be transitive. A homogeneous relation R on the set X is a transitive relation if, for all a, b, c ∈ X, if a R b and b R c, then a R c. Or in terms of first-order logic: where a R b is the infix notation for (a, b) ∈ R. As a non-mathematical example, the relation "is an ancestor of" is transitive.
Early childhood traumaEarly childhood trauma refers to various types of adversity and traumatic events experienced during the early years of a person's life. This is deemed the most critical developmental period in human life by psychologists. A critical period refers to a sensitive time during the early years of childhood in which children may be more vulnerable to be affected by environmental stimulation. These traumatic events can include serious sickness, natural disasters, family violence, sudden separation from a family member, being the victim of abuse, or suffering the loss of a loved one.
Profinite groupIn mathematics, a profinite group is a topological group that is in a certain sense assembled from a system of finite groups. The idea of using a profinite group is to provide a "uniform", or "synoptic", view of an entire system of finite groups. Properties of the profinite group are generally speaking uniform properties of the system. For example, the profinite group is finitely generated (as a topological group) if and only if there exists such that every group in the system can be generated by elements.
