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58-411: The isthmus is a roughly trapezoidal region of low-lying swampy terrain, about 15 km long, 24 km wide at the southeast (mainland) end, and 8 km wide at the northwest ("island") end. Both ends are limited by mountainous terrain. The northeast and southwest shores are smooth and gently curved inward, about 11 and 20 km long, respectively. The southwest shore of the isthmus ends next to

116-406: A = 0 {\displaystyle d-c=b-a=0} , but it is an ex-tangential quadrilateral (which is not a trapezoid) when | d − c | = | b − a | ≠ 0 {\displaystyle |d-c|=|b-a|\neq 0} . Given a convex quadrilateral, the following properties are equivalent, and each implies that the quadrilateral

174-543: A and b , if a = b , then a ≥ 0 implies b ≥ 0 (here, ϕ ( x ) {\displaystyle \phi (x)} is x ≥ 0 ) These properties offer a formal reinterpretation of equality from how it is defined in standard Zermelo–Fraenkel set theory (ZFC) or other formal foundations . In ZFC, equality only means that two sets have the same elements. However, outside of set theory , mathematicians don't tend to view their objects of interest as sets. For instance, many mathematicians would say that

232-569: A complete axiomatization of equality, meaning, if they were to define equality, then the converse of the second statement must be true. The converse of the Substitution property is the identity of indiscernibles , which states that two distinct things cannot have all their properties in common. In mathematics, the identity of indiscernibles is usually rejected since indiscernibles in mathematical logic are not necessarily forbidden. Set equality in ZFC

290-418: A special case the well-known formula for the area of a triangle , by considering a triangle as a degenerate trapezoid in which one of the parallel sides has shrunk to a point. The 7th-century Indian mathematician Bhāskara I derived the following formula for the area of a trapezoid with consecutive sides a , c , b , d : where a and b are parallel and b > a . This formula can be factored into

348-425: A choice – any statement that identifies them "depends on choice of identification". This distinction, between equality and isomorphism , is of fundamental importance in category theory and is one motivation for the development of category theory. In some cases, one may consider as equal two mathematical objects that are only equivalent for the properties and structure being considered. The word congruence (and

406-474: A complete axiomatization. However, apart from cases dealing with indiscernibles, these properties taken as axioms of equality are equivalent to equality as defined in ZFC. These are sometimes taken as the definition of equality, such as in some areas of first-order logic . The Law of identity is distinct from reflexivity in two main ways: first, the Law of Identity applies only to cases of equality, and second, it

464-401: A more symmetric version When one of the parallel sides has shrunk to a point (say a = 0), this formula reduces to Heron's formula for the area of a triangle. Another equivalent formula for the area, which more closely resembles Heron's formula, is where s = 1 2 ( a + b + c + d ) {\displaystyle s={\tfrac {1}{2}}(a+b+c+d)}

522-446: A set: those binary relations that are reflexive , symmetric and transitive . The identity relation is an equivalence relation. Conversely, let R be an equivalence relation, and let us denote by x the equivalence class of x , consisting of all elements z such that x R z . Then the relation x R y is equivalent with the equality x  =  y . It follows that equality is the finest equivalence relation on any set S in

580-408: A single output based on a select signal. Typical designs will employ trapezoids without specifically stating they are multiplexors as they are universally equivalent. Equality (mathematics) In mathematics , equality is a relationship between two quantities or expressions , stating that they have the same value, or represent the same mathematical object . Equality between A and B

638-523: A transposition of the terms. This was reversed in British English in about 1875, but it has been retained in American English to the present. The following table compares usages, with the most specific definitions at the top to the most general at the bottom. There is some disagreement whether parallelograms , which have two pairs of parallel sides, should be regarded as trapezoids. Some define

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696-438: A trapezoid as a quadrilateral having only one pair of parallel sides (the exclusive definition), thereby excluding parallelograms. Some sources use the term proper trapezoid to describe trapezoids under the exclusive definition, analogous to uses of the word proper in some other mathematical objects. Others define a trapezoid as a quadrilateral with at least one pair of parallel sides (the inclusive definition ), making

754-479: A trapezoid is given by where a and b are the lengths of the parallel sides, h is the height (the perpendicular distance between these sides), and m is the arithmetic mean of the lengths of the two parallel sides. In 499 AD Aryabhata , a great mathematician - astronomer from the classical age of Indian mathematics and Indian astronomy , used this method in the Aryabhatiya (section 2.8). This yields as

812-447: Is ( x + 1 ) ( x + 1 ) = x 2 + 2 x + 1 {\displaystyle \left(x+1\right)\left(x+1\right)=x^{2}+2x+1} is true for all real numbers x {\displaystyle x} . There is no standard notation that distinguishes an equation from an identity, or other use of the equality relation: one has to guess an appropriate interpretation from

870-423: Is a quadrilateral that has one pair of parallel sides. The parallel sides are called the bases of the trapezoid. The other two sides are called the legs (or the lateral sides ) if they are not parallel; otherwise, the trapezoid is a parallelogram, and there are two pairs of bases. A scalene trapezoid is a trapezoid with no sides of equal measure, in contrast with the special cases below. A trapezoid

928-441: Is a trapezoid: Additionally, the following properties are equivalent, and each implies that opposite sides a and b are parallel: The midsegment of a trapezoid is the segment that joins the midpoints of the legs. It is parallel to the bases. Its length m is equal to the average of the lengths of the bases a and b of the trapezoid, The midsegment of a trapezoid is one of the two bimedians (the other bimedian divides

986-467: Is an unusual alternation of sand bars and coastal marshes, showing species of mountainous and steppe vegetation growing side by side. Some trees like bird cherry and common pines grow close to ground, as creeping bush. Sand levees stretch for many kilometers along the banks of the isthmus. The isthmus is almost divided in two by a shallow body of water, Lake Arangatuy (or Bol'shoy Sor), measuring about 13 by 7 km and about 54 km of area. The lake

1044-426: Is capable of declairing these indiscernibles as not equal, but an equality solely defined by these properties is not. Thus these properties form a strictly weaker notion of equality than set equality in ZFC. Outside of pure math , the identity of indiscernibles has attracted much controversy and criticism, especially from corpuscular philosophy and quantum mechanics . This is why the properties are said to not form

1102-404: Is fed on the eastern side by streams from the mainland, and its outlet is near the north corner of the isthmus, near the "island", just east of the small village of Monakhovo - Zmeyevaya . The lake and its bays are inhabited by dace, perch, pike, and other types of fish. Many rare bird species nest on its shores: whooper swan, black-throated loon, Eurasian curlew, and others. The southwest side of

1160-441: Is intentional. This makes it an incomplete axiomatization of equality. That is, it does not say what equality is , only what "equality" must satify. However, the two axioms as stated are still generally useful, even as an incomplete axiomatization of equality, as they are usually sufficient for deducing most properties of equality that mathematicians care about. (See the following subsection) If these properties were to define

1218-453: Is not restricted to elements of a set. However, many mathematicians refer to both as "Reflexivity", which is generally harmless. This is also sometimes included in the axioms of equality, but isn't necessary as it can be deduced from the other two axioms as shown above. There are some logic systems that do not have any notion of equality. This reflects the undecidability of the equality of two real numbers , defined by formulas involving

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1276-407: Is not transitive (since many small differences can add up to something big). However, equality almost everywhere is transitive. A questionable equality under test may be denoted using the = ? {\displaystyle {\stackrel {?}{=}}} symbol . Viewed as a relation , equality is the archetype of the more general concept of an equivalence relation on

1334-431: Is possible for acute trapezoids or right trapezoids (as rectangles). A parallelogram is (under the inclusive definition) a trapezoid with two pairs of parallel sides. A parallelogram has central 2-fold rotational symmetry (or point reflection symmetry). It is possible for obtuse trapezoids or right trapezoids (rectangles). A tangential trapezoid is a trapezoid that has an incircle . A Saccheri quadrilateral

1392-493: Is similar to a trapezoid in the hyperbolic plane, with two adjacent right angles, while it is a rectangle in the Euclidean plane . A Lambert quadrilateral in the hyperbolic plane has 3 right angles. Four lengths a , c , b , d can constitute the consecutive sides of a non-parallelogram trapezoid with a and b parallel only when The quadrilateral is a parallelogram when d − c = b −

1450-410: Is the semiperimeter of the trapezoid. (This formula is similar to Brahmagupta's formula , but it differs from it, in that a trapezoid might not be cyclic (inscribed in a circle). The formula is also a special case of Bretschneider's formula for a general quadrilateral ). From Bretschneider's formula, it follows that The bimedian connecting the parallel sides bisects the area. The lengths of

1508-403: Is the problem of finding values of some variable, called unknown , for which the specified equality is true. Each value of the unknown for which the equation holds is called a solution of the given equation; also stated as satisfying the equation. For example, the equation x 2 − 6 x + 5 = 0 {\displaystyle x^{2}-6x+5=0} has

1566-410: Is the unique equivalence relation on S {\displaystyle S} whose equivalence classes are all singletons . Given operations over S {\displaystyle S} , that last property makes equality a congruence relation . In logic , a predicate is a proposition which may have some free variables . Equality is a predicate, which may be true for some values of

1624-487: Is usually considered to be a convex quadrilateral in Euclidean geometry , but there are also crossed cases. If ABCD is a convex trapezoid, then ABDC is a crossed trapezoid. The metric formulas in this article apply in convex trapezoids. The ancient Greek mathematician Euclid defined five types of quadrilateral, of which four had two sets of parallel sides (known in English as square, rectangle, rhombus and rhomboid) and

1682-421: Is written A  =  B , and pronounced " A equals B ". In this equality, A and B are distinguished by calling them left-hand side ( LHS ), and right-hand side ( RHS ). Two objects that are not equal are said to be distinct . A formula such as x = y , {\displaystyle x=y,} where x and y are any expressions, means that x and y denote or represent

1740-417: The integers , the basic arithmetic operations , the logarithm and the exponential function . In other words, there cannot exist any algorithm for deciding such an equality (see Richardson's theorem ). The binary relation " is approximately equal " (denoted by the symbol ≈ {\displaystyle \approx } ) between real numbers or other things, even if more precisely defined,

1798-414: The trapezoidal rule for estimating areas under a curve. An acute trapezoid has two adjacent acute angles on its longer base edge. An obtuse trapezoid on the other hand has one acute and one obtuse angle on each base . An isosceles trapezoid is a trapezoid where the base angles have the same measure. As a consequence the two legs are also of equal length and it has reflection symmetry . This

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1856-416: The unit circle in analytic geometry ; therefore, this equation is called the equation of the unit circle . See also: Equation solving An identity is an equality that is true for all values of its variables in a given domain. An "equation" may sometimes mean an identity, but more often than not, it specifies a subset of the variable space to be the subset where the equation is true. An example

1914-452: The angle bisectors to angles A and B intersect at P , and the angle bisectors to angles C and D intersect at Q , then In architecture the word is used to refer to symmetrical doors, windows, and buildings built wider at the base, tapering toward the top, in Egyptian style. If these have straight sides and sharp angular corners, their shapes are usually isosceles trapezoids . This was

1972-482: The associated symbol ≅ {\displaystyle \cong } ) is frequently used for this kind of equality, and is defined as the quotient set of the isomorphism classes between the objects. In geometry for instance, two geometric shapes are said to be equal or congruent when one may be moved to coincide with the other, and the equality/congruence relation is the isomorphism classes of isometries between shapes. Similarly to isomorphisms of sets,

2030-417: The axiom of extensionality states that two sets which contain the same elements are the same set. Incorporating half of the work into the first-order logic may be regarded as a mere matter of convenience, as noted by Lévy. In first-order logic without equality, two sets are defined to be equal if they contain the same elements. Then the axiom of extensionality states that two equal sets are contained in

2088-420: The diagonals are where a is the short base, b is the long base, and c and d are the trapezoid legs. If the trapezoid is divided into four triangles by its diagonals AC and BD (as shown on the right), intersecting at O , then the area of △ {\displaystyle \triangle } AOD is equal to that of △ {\displaystyle \triangle } BOC , and

2146-457: The difference between isomorphisms and equality/congruence between such mathematical objects with properties and structure was one motivation for the development of category theory , as well as for homotopy type theory and univalent foundations . Equality of sets is axiomatized in set theory in two different ways, depending on whether the axioms are based on a first-order language with or without equality. In first-order logic with equality,

2204-468: The expression " 1 ∪ 2 {\displaystyle 1\cup 2} " (see union ) is an abuse of notation or meaningless. This is a more abstracted framework which can be grounded in ZFC (that is, both axioms can be proved within ZFC as well as most other formal foundations), but is closer to how most mathematicians use equality. Note that this says "Equality implies these two properties" not that "These properties define equality"; this

2262-403: The extended nonparallel sides and the intersection point of the diagonals, bisects each base. The center of area (center of mass for a uniform lamina ) lies along the line segment joining the midpoints of the parallel sides, at a perpendicular distance x from the longer side b given by The center of area divides this segment in the ratio (when taken from the short to the long side) If

2320-401: The following properties: ∀ a ( a = a ) {\displaystyle \forall a(a=a)} ( a = b ) ⟹ [ ϕ ( a ) ⇒ ϕ ( b ) ] {\displaystyle (a=b)\implies {\bigl [}\phi (a)\Rightarrow \phi (b){\bigr ]}} For example: For all real numbers

2378-802: The isthmus is a beach of exceptionally clean sand, Myagkaya Karga . The road leading to the Svyatoi Nos crosses the isthmus parallel to the beach. The Kulina marshes have about 120 mud volcanoes ( gryphons ) and hydrothermal springs, on land and underwater, scattered over about 40 km. The waters may be up to 80 °C, and contain high levels of dissolved salts and other chemicals, up to 3 g / L – chiefly sodium sulfate , chloride , and fluoride , as well as silicic acid . The gryphons range in diameter from 20 cm to 7 m, and are responsible for many small shallow warm brackish ponds, round or oval, with depths ranging from 0.5 to 5.0 m, and areas from 10 to 300 m, whose level may be up to 1 meter above or below

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2436-519: The last did not have two sets of parallel sides – a τραπέζια ( trapezia literally 'table', itself from τετράς ( tetrás ) 'four' + πέζα ( péza ) 'foot; end, border, edge'). Two types of trapezia were introduced by Proclus (AD 412 to 485) in his commentary on the first book of Euclid's Elements : All European languages follow Proclus's structure as did English until the late 18th century, until an influential mathematical dictionary published by Charles Hutton in 1795 supported without explanation

2494-399: The lengths of the parallel sides. Let the trapezoid have vertices A , B , C , and D in sequence and have parallel sides AB and DC . Let E be the intersection of the diagonals, and let F be on side DA and G be on side BC such that FEG is parallel to AB and CD . Then FG is the harmonic mean of AB and DC : The line that goes through both the intersection point of

2552-486: The level of Lake Baikal. The largest ones cover up to 2500 m. The springs also form wam rivers that rarely or never freeze in winter. The main group of those hydrothermal-fed lakes, which includes Lake Bormashov , near the mouth of the Barguzin river. The waters and bottom mud ( sapropel ) are reputed to have health properties. Until recently, the settlement Kulinoe, Buryatia was located near an active mud volcano. Due to

2610-446: The members are interpreted as numbers or sets, but are false if the members are interpreted as expressions or sequences of symbols. An identity , such as ( x + 1 ) 2 = x 2 + 2 x + 1 , {\displaystyle (x+1)^{2}=x^{2}+2x+1,} means that if x is replaced with any number, then the two expressions take the same value. This may also be interpreted as saying that

2668-716: The parallelogram a special type of trapezoid. The latter definition is consistent with its uses in higher mathematics such as calculus . This article uses the inclusive definition and considers parallelograms as special cases of a trapezoid. This is also advocated in the taxonomy of quadrilaterals . Under the inclusive definition, all parallelograms (including rhombuses , squares and non-square rectangles ) are trapezoids. Rectangles have mirror symmetry on mid-edges; rhombuses have mirror symmetry on vertices, while squares have mirror symmetry on both mid-edges and vertices. A right trapezoid (also called right-angled trapezoid ) has two adjacent right angles . Right trapezoids are used in

2726-422: The product of the areas of △ {\displaystyle \triangle } AOD and △ {\displaystyle \triangle } BOC is equal to that of △ {\displaystyle \triangle } AOB and △ {\displaystyle \triangle } COD . The ratio of the areas of each pair of adjacent triangles is the same as that between

2784-418: The same object. For example, are two notations for the same number. Similarly, using set builder notation , since the two sets have the same elements. (This equality results from the axiom of extensionality that is often expressed as "two sets that have the same elements are equal". ) The truth of an equality depends on an interpretation of its members. In the above examples, the equalities are true if

2842-478: The same rational number (the same point on a number line). This distinction gives rise to the notion of a quotient set . Similarly, the sets are not equal sets – the first consists of letters, while the second consists of numbers – but they are both sets of three elements and thus isomorphic, meaning that there is a bijection between them. For example However, there are other choices of isomorphism, such as and these sets cannot be identified without making such

2900-424: The semantics of expressions and the context. Sometimes, but not always, an identity is written with a triple bar : ( x + 1 ) ( x + 1 ) ≡ x 2 + 2 x + 1. {\displaystyle \left(x+1\right)\left(x+1\right)\equiv x^{2}+2x+1.} In mathematical logic and mathematical philosophy , equality is often described through

2958-565: The sense that it is the relation that has the smallest equivalence classes (every class is reduced to a single element). In some contexts, equality is sharply distinguished from equivalence or isomorphism . For example, one may distinguish fractions from rational numbers , the latter being equivalence classes of fractions: the fractions 1 / 2 {\displaystyle 1/2} and 2 / 4 {\displaystyle 2/4} are distinct as fractions (as different strings of symbols) but they "represent"

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3016-776: The standard style for the doors and windows of the Inca . The crossed ladders problem is the problem of finding the distance between the parallel sides of a right trapezoid, given the diagonal lengths and the distance from the perpendicular leg to the diagonal intersection. In morphology , taxonomy and other descriptive disciplines in which a term for such shapes is necessary, terms such as trapezoidal or trapeziform commonly are useful in descriptions of particular organs or forms. In computer engineering, specifically digital logic and computer architecture, trapezoids are typically utilized to symbolize multiplexors . Multiplexors are logic elements that select between multiple elements and produce

3074-536: The threat of gas poisoning and livestock poisoning by the salty water, residents were forced to relocate. 53°33′54″N 109°01′16″E  /  53.565°N 109.021°E  / 53.565; 109.021 Trapezoid In geometry , a trapezoid ( / ˈ t r æ p ə z ɔɪ d / ) in North American English , or trapezium ( / t r ə ˈ p iː z i ə m / ) in British English ,

3132-489: The town of Ust-Barguzin . The isthmus divides the strait between the island and the mainland into two bays, Chivyrkuisky Bay at the northeast and Barguzinsky Bay at the southwest. Thousands of years ago the isthmus did not exist, and Svyatoi Nos was an island. It was created by alluvial sediments of the Barguzin River. Chivyrkuisky Isthmus is one of the three nesting areas of Baikal's waterfowl and birds of prey. It

3190-443: The trapezoid into equal areas). The height (or altitude) is the perpendicular distance between the bases. In the case that the two bases have different lengths ( a ≠ b ), the height of a trapezoid h can be determined by the length of its four sides using the formula where c and d are the lengths of the legs and p = a + b + c + d {\displaystyle p=a+b+c+d} . The area K of

3248-620: The two sides of the equals sign represent the same function (equality of functions), or that the two expressions denote the same polynomial (equality of polynomials). The word is derived from the Latin aequālis ("equal", "like", "comparable", "similar"), which itself stems from aequus ("equal", "level", "fair", "just"). If restricted to the elements of a given set S {\displaystyle S} , those first three properties make equality an equivalence relation on S {\displaystyle S} . In fact, equality

3306-507: The values x = 1 {\displaystyle x=1} and x = 5 {\displaystyle x=5} as its only solutions. The terminology is used similarly for equations with several unknowns. An equation can be used to define a set. For example, the set of all solution pairs ( x , y ) {\displaystyle (x,y)} of the equation x 2 + y 2 = 1 {\displaystyle x^{2}+y^{2}=1} forms

3364-423: The variables (if any) and false for other values. More specifically, equality is a binary relation (i.e., a two-argument predicate) which may produce a truth value ( true or false ) from its arguments. In computer programming , equality is called a Boolean -valued expression , and its computation from the two expressions is known as comparison . See also: Relational operator § Equality An equation

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