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57-449: In many scenarios, the use of "without loss of generality" is made possible by the presence of symmetry . For example, if some property P ( x , y ) of real numbers is known to be symmetric in x and y , namely that P ( x , y ) is equivalent to P ( y , x ), then in proving that P ( x , y ) holds for every x and y , one may assume "without loss of generality" that x ≤ y . There is no loss of generality in this assumption, since once

114-435: A logical fallacy of proving a claim by proving a non-representative example. Consider the following theorem (which is a case of the pigeonhole principle ): If three objects are each painted either red or blue, then there must be at least two objects of the same color. A proof: Assume, without loss of generality, that the first object is red. If either of the other two objects is red, then we are finished; if not, then

171-413: A Boolean semantic. For example, lazy evaluation is sometimes implemented for P  ∧  Q and P  ∨  Q , so these connectives are not commutative if either or both of the expressions P , Q have side effects . Also, a conditional , which in some sense corresponds to the material conditional connective, is essentially non-Boolean because for if (P) then Q; , the consequent Q is not executed if

228-655: A block) with each smaller piece usually consisting of fabric triangles, the craft lends itself readily to the application of symmetry. Symmetries appear in the design of objects of all kinds. Examples include beadwork , furniture , sand paintings , knotwork , masks , and musical instruments . Symmetries are central to the art of M.C. Escher and the many applications of tessellation in art and craft forms such as wallpaper , ceramic tilework such as in Islamic geometric decoration , batik , ikat , carpet-making, and many kinds of textile and embroidery patterns. Symmetry

285-523: A classical compositional semantics with a robust pragmatics . A logical connective is similar to, but not equivalent to, a syntax commonly used in programming languages called a conditional operator . In formal languages , truth functions are represented by unambiguous symbols. This allows logical statements to not be understood in an ambiguous way. These symbols are called logical connectives , logical operators , propositional operators , or, in classical logic , truth-functional connectives . For

342-411: A list of journals and newsletters known to deal, at least in part, with symmetry and the arts. Symmetry finds its ways into architecture at every scale, from the overall external views of buildings such as Gothic cathedrals and The White House , through the layout of the individual floor plans , and down to the design of individual building elements such as tile mosaics . Islamic buildings such as

399-402: A logical connective as converse implication " ← {\displaystyle \leftarrow } " is actually the same as material conditional with swapped arguments; thus, the symbol for converse implication is redundant. In some logical calculi (notably, in classical logic ), certain essentially different compound statements are logically equivalent . A less trivial example of

456-522: A redundancy is the classical equivalence between ¬ p ∨ q {\displaystyle \neg p\vee q} and p → q {\displaystyle p\to q} . Therefore, a classical-based logical system does not need the conditional operator " → {\displaystyle \to } " if " ¬ {\displaystyle \neg } " (not) and " ∨ {\displaystyle \vee } " (or) are already in use, or may use

513-479: A sense of harmonious and beautiful proportion and balance. In mathematics , the term has a more precise definition and is usually used to refer to an object that is invariant under some transformations , such as translation , reflection , rotation , or scaling . Although these two meanings of the word can sometimes be told apart, they are intricately related, and hence are discussed together in this article. Mathematical symmetry may be observed with respect to

570-467: A theorem. The situation, however, is more complicated in intuitionistic logic . Of its five connectives, {∧, ∨, →, ¬, ⊥}, only negation "¬" can be reduced to other connectives (see False (logic) § False, negation and contradiction for more). Neither conjunction, disjunction, nor material conditional has an equivalent form constructed from the other four logical connectives. The standard logical connectives of classical logic have rough equivalents in

627-405: A variety of alternative interpretations in nonclassical logics . Their classical interpretations are similar to the meanings of natural language expressions such as English "not", "or", "and", and "if", but not identical. Discrepancies between natural language connectives and those of classical logic have motivated nonclassical approaches to natural language meaning as well as approaches which pair

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684-479: Is nonclassical . However, others maintain classical semantics by positing pragmatic accounts of exclusivity which create the illusion of nonclassicality. In such accounts, exclusivity is typically treated as a scalar implicature . Related puzzles involving disjunction include free choice inferences , Hurford's Constraint , and the contribution of disjunction in alternative questions . Other apparent discrepancies between natural language and classical logic include

741-463: Is a consequence of corresponding "...→..." connectives for propositional variables. Some many-valued logics may have incompatible definitions of equivalence and order (entailment). Both conjunction and disjunction are associative, commutative and idempotent in classical logic, most varieties of many-valued logic and intuitionistic logic. The same is true about distributivity of conjunction over disjunction and disjunction over conjunction, as well as for

798-526: Is a corresponding conserved quantity such as energy or momentum; a conserved current, in Noether's original language); and also, Wigner's classification , which says that the symmetries of the laws of physics determine the properties of the particles found in nature. Important symmetries in physics include continuous symmetries and discrete symmetries of spacetime ; internal symmetries of particles; and supersymmetry of physical theories. In biology,

855-442: Is also used in designing logos. By creating a logo on a grid and using the theory of symmetry, designers can organize their work, create a symmetric or asymmetrical design, determine the space between letters, determine how much negative space is required in the design, and how to accentuate parts of the logo to make it stand out. Symmetry is not restricted to the visual arts. Its role in the history of music touches many aspects of

912-541: Is faster when this is a property of a single object. Studies of human perception and psychophysics have shown that detection of symmetry is fast, efficient and robust to perturbations. For example, symmetry can be detected with presentations between 100 and 150 milliseconds. More recent neuroimaging studies have documented which brain regions are active during perception of symmetry. Sasaki et al. used functional magnetic resonance imaging (fMRI) to compare responses for patterns with symmetrical or random dots. A strong activity

969-478: Is implemented as logic gates in digital circuits . Practically all digital circuits (the major exception is DRAM ) are built up from NAND , NOR , NOT , and transmission gates ; see more details in Truth function in computer science . Logical operators over bit vectors (corresponding to finite Boolean algebras ) are bitwise operations . But not every usage of a logical connective in computer programming has

1026-503: Is short for ( P ∨ ( Q ∧ ( ¬ R ) ) ) → S {\displaystyle (P\vee (Q\wedge (\neg R)))\rightarrow S} . Here is a table that shows a commonly used precedence of logical operators. However, not all compilers use the same order; for instance, an ordering in which disjunction is lower precedence than implication or bi-implication has also been used. Sometimes precedence between conjunction and disjunction

1083-412: Is the tendency for excessive symmetry to be perceived as boring or uninteresting. Rudolf Arnheim suggested that people prefer shapes that have some symmetry, and enough complexity to make them interesting. Symmetry can be found in various forms in literature , a simple example being the palindrome where a brief text reads the same forwards or backwards. Stories may have a symmetrical structure, such as

1140-526: Is to choose a minimal set, and define other connectives by some logical form, as in the example with the material conditional above. The following are the minimal functionally complete sets of operators in classical logic whose arities do not exceed 2: Another approach is to use with equal rights connectives of a certain convenient and functionally complete, but not minimal set. This approach requires more propositional axioms , and each equivalence between logical forms must be either an axiom or provable as

1197-684: Is unspecified requiring to provide it explicitly in given formula with parentheses. The order of precedence determines which connective is the "main connective" when interpreting a non-atomic formula. The 16 logical connectives can be partially ordered to produce the following Hasse diagram . The partial order is defined by declaring that x ≤ y {\displaystyle x\leq y} if and only if whenever x {\displaystyle x} holds then so does y . {\displaystyle y.} Logical connectives are used in computer science and in set theory . A truth-functional approach to logical operators

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1254-674: The Taj Mahal and the Lotfollah mosque make elaborate use of symmetry both in their structure and in their ornamentation. Moorish buildings like the Alhambra are ornamented with complex patterns made using translational and reflection symmetries as well as rotations. It has been said that only bad architects rely on a "symmetrical layout of blocks, masses and structures"; Modernist architecture , starting with International style , relies instead on "wings and balance of masses". Since

1311-613: The diatonic scale or the major chord . Symmetrical scales or chords, such as the whole tone scale , augmented chord , or diminished seventh chord (diminished-diminished seventh), are said to lack direction or a sense of forward motion, are ambiguous as to the key or tonal center, and have a less specific diatonic functionality . However, composers such as Alban Berg , Béla Bartók , and George Perle have used axes of symmetry and/or interval cycles in an analogous way to keys or non- tonal tonal centers . George Perle explains that "C–E, D–F♯, [and] Eb–G, are different instances of

1368-489: The moral message "we are all the same" while asymmetrical interactions may send the message "I am special; better than you." Peer relationships, such as can be governed by the Golden Rule , are based on symmetry, whereas power relationships are based on asymmetry. Symmetrical relationships can to some degree be maintained by simple ( game theory ) strategies seen in symmetric games such as tit for tat . There exists

1425-435: The paradoxes of material implication , donkey anaphora and the problem of counterfactual conditionals . These phenomena have been taken as motivation for identifying the denotations of natural language conditionals with logical operators including the strict conditional , the variably strict conditional , as well as various dynamic operators. The following table shows the standard classically definable approximations for

1482-573: The " → {\displaystyle \to } " only as a syntactic sugar for a compound having one negation and one disjunction. There are sixteen Boolean functions associating the input truth values p {\displaystyle p} and q {\displaystyle q} with four-digit binary outputs. These correspond to possible choices of binary logical connectives for classical logic . Different implementations of classical logic can choose different functionally complete subsets of connectives. One approach

1539-490: The English connectives. Some logical connectives possess properties that may be expressed in the theorems containing the connective. Some of those properties that a logical connective may have are: For classical and intuitionistic logic, the "=" symbol means that corresponding implications "...→..." and "...←..." for logical compounds can be both proved as theorems, and the "≤" symbol means that "...→..." for logical compounds

1596-557: The absorption law. In classical logic and some varieties of many-valued logic, conjunction and disjunction are dual, and negation is self-dual, the latter is also self-dual in intuitionistic logic. As a way of reducing the number of necessary parentheses, one may introduce precedence rules : ¬ has higher precedence than ∧, ∧ higher than ∨, and ∨ higher than →. So for example, P ∨ Q ∧ ¬ R → S {\displaystyle P\vee Q\wedge {\neg R}\rightarrow S}

1653-409: The arts, covering architecture , art , and music. The opposite of symmetry is asymmetry , which refers to the absence of symmetry. A geometric shape or object is symmetric if it can be divided into two or more identical pieces that are arranged in an organized fashion. This means that an object is symmetric if there is a transformation that moves individual pieces of the object, but doesn't change

1710-401: The body becomes bilaterally symmetric for the purpose of movement, with symmetrical pairs of muscles and skeletal elements, though internal organs often remain asymmetric. Plants and sessile (attached) animals such as sea anemones often have radial or rotational symmetry , which suits them because food or threats may arrive from any direction. Fivefold symmetry is found in the echinoderms ,

1767-456: The case x ≤ y ⇒ P ( x , y ) has been proved, the other case follows by interchanging x and y  : y ≤ x ⇒ P ( y , x ), and by symmetry of P , this implies P ( x , y ), thereby showing that P ( x , y ) holds for all cases. On the other hand, if neither such a symmetry nor another form of equivalence can be established, then the use of "without loss of generality" is incorrect and can amount to an instance of proof by example –

Without loss of generality - Misplaced Pages Continue

1824-421: The connective if (→) is not symmetric. Other symmetric logical connectives include nand (not-and, or ⊼), xor (not-biconditional, or ⊻), and nor (not-or, or ⊽). Generalizing from geometrical symmetry in the previous section, one can say that a mathematical object is symmetric with respect to a given mathematical operation , if, when applied to the object, this operation preserves some property of

1881-487: The creation and perception of music. Symmetry has been used as a formal constraint by many composers, such as the arch (swell) form (ABCBA) used by Steve Reich , Béla Bartók , and James Tenney . In classical music, Johann Sebastian Bach used the symmetry concepts of permutation and invariance. Symmetry is also an important consideration in the formation of scales and chords , traditional or tonal music being made up of non-symmetrical groups of pitches , such as

1938-531: The cycle of fourths) will produce the diatonic major scale. Cyclic tonal progressions in the works of Romantic composers such as Gustav Mahler and Richard Wagner form a link with the cyclic pitch successions in the atonal music of Modernists such as Bartók, Alexander Scriabin , Edgard Varèse , and the Vienna school. At the same time, these progressions signal the end of tonality. The first extended composition consistently based on symmetrical pitch relations

1995-471: The earliest uses of pottery wheels to help shape clay vessels, pottery has had a strong relationship to symmetry. Pottery created using a wheel acquires full rotational symmetry in its cross-section, while allowing substantial freedom of shape in the vertical direction. Upon this inherently symmetrical starting point, potters from ancient times onwards have added patterns that modify the rotational symmetry to achieve visual objectives. Cast metal vessels lacked

2052-453: The following ones. For example, the meaning of the statements it is raining (denoted by p {\displaystyle p} ) and I am indoors (denoted by q {\displaystyle q} ) is transformed, when the two are combined with logical connectives: It is also common to consider the always true formula and the always false formula to be connective (in which case they are nullary ). This table summarizes

2109-686: The grammars of natural languages. In English , as in many languages, such expressions are typically grammatical conjunctions . However, they can also take the form of complementizers , verb suffixes , and particles . The denotations of natural language connectives is a major topic of research in formal semantics , a field that studies the logical structure of natural languages. The meanings of natural language connectives are not precisely identical to their nearest equivalents in classical logic. In particular, disjunction can receive an exclusive interpretation in many languages. Some researchers have taken this fact as evidence that natural language semantics

2166-505: The group that includes starfish , sea urchins , and sea lilies . In biology, the notion of symmetry is also used as in physics, that is to say to describe the properties of the objects studied, including their interactions. A remarkable property of biological evolution is the changes of symmetry corresponding to the appearance of new parts and dynamics. Symmetry is important to chemistry because it undergirds essentially all specific interactions between molecules in nature (i.e., via

2223-413: The inherent rotational symmetry of wheel-made pottery, but otherwise provided a similar opportunity to decorate their surfaces with patterns pleasing to those who used them. The ancient Chinese , for example, used symmetrical patterns in their bronze castings as early as the 17th century BC. Bronze vessels exhibited both a bilateral main motif and a repetitive translated border design. A long tradition of

2280-548: The interaction of natural and human-made chiral molecules with inherently chiral biological systems). The control of the symmetry of molecules produced in modern chemical synthesis contributes to the ability of scientists to offer therapeutic interventions with minimal side effects . A rigorous understanding of symmetry explains fundamental observations in quantum chemistry , and in the applied areas of spectroscopy and crystallography . The theory and application of symmetry to these areas of physical science draws heavily on

2337-470: The mathematical area of group theory . For a human observer, some symmetry types are more salient than others, in particular the most salient is a reflection with a vertical axis, like that present in the human face. Ernst Mach made this observation in his book "The analysis of sensations" (1897), and this implies that perception of symmetry is not a general response to all types of regularities. Both behavioural and neurophysiological studies have confirmed

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2394-526: The most powerful tools of theoretical physics , as it has become evident that practically all laws of nature originate in symmetries. In fact, this role inspired the Nobel laureate PW Anderson to write in his widely read 1972 article More is Different that "it is only slightly overstating the case to say that physics is the study of symmetry." See Noether's theorem (which, in greatly simplified form, states that for every continuous mathematical symmetry, there

2451-423: The notion of symmetry is mostly used explicitly to describe body shapes. Bilateral animals , including humans, are more or less symmetric with respect to the sagittal plane which divides the body into left and right halves. Animals that move in one direction necessarily have upper and lower sides, head and tail ends, and therefore a left and a right. The head becomes specialized with a mouth and sense organs, and

2508-757: The object. The set of operations that preserve a given property of the object form a group . In general, every kind of structure in mathematics will have its own kind of symmetry. Examples include even and odd functions in calculus , symmetric groups in abstract algebra , symmetric matrices in linear algebra , and Galois groups in Galois theory . In statistics , symmetry also manifests as symmetric probability distributions , and as skewness —the asymmetry of distributions. Symmetry in physics has been generalized to mean invariance —that is, lack of change—under any kind of transformation, for example arbitrary coordinate transformations . This concept has become one of

2565-585: The other two objects must both be blue and we are still finished. The above argument works because the exact same reasoning could be applied if the alternative assumption, namely, that the first object is blue, were made, or, similarly, that the words 'red' and 'blue' can be freely exchanged in the wording of the proof. As a result, the use of "without loss of generality" is valid in this case. Symmetry Symmetry (from Ancient Greek συμμετρία ( summetría )  'agreement in dimensions, due proportion, arrangement') in everyday life refers to

2622-546: The overall shape. The type of symmetry is determined by the way the pieces are organized, or by the type of transformation: A dyadic relation R = S × S is symmetric if for all elements a , b in S , whenever it is true that Rab , it is also true that Rba . Thus, the relation "is the same age as" is symmetric, for if Paul is the same age as Mary, then Mary is the same age as Paul. In propositional logic, symmetric binary logical connectives include and (∧, or &), or (∨, or |) and if and only if (↔), while

2679-421: The passage of time ; as a spatial relationship ; through geometric transformations ; through other kinds of functional transformations; and as an aspect of abstract objects , including theoretic models , language , and music . This article describes symmetry from three perspectives: in mathematics , including geometry , the most familiar type of symmetry for many people; in science and nature ; and in

2736-428: The rise and fall pattern of Beowulf . Logical connective In logic , a logical connective (also called a logical operator , sentential connective , or sentential operator ) is a logical constant . Connectives can be used to connect logical formulas. For instance in the syntax of propositional logic , the binary connective ∨ {\displaystyle \lor } can be used to join

2793-459: The rules which allow new well-formed formulas to be constructed by joining other well-formed formulas using truth-functional connectives, see well-formed formula . Logical connectives can be used to link zero or more statements, so one can speak about n -ary logical connectives . The boolean constants True and False can be thought of as zero-ary operators. Negation is a 1-ary connective, and so on. Commonly used logical connectives include

2850-417: The same interval … the other kind of identity. … has to do with axes of symmetry. C–E belongs to a family of symmetrically related dyads as follows:" Thus in addition to being part of the interval-4 family, C–E is also a part of the sum-4 family (with C equal to 0). Interval cycles are symmetrical and thus non-diatonic. However, a seven pitch segment of C5 (the cycle of fifths, which are enharmonic with

2907-507: The special sensitivity to reflection symmetry in humans and also in other animals. Early studies within the Gestalt tradition suggested that bilateral symmetry was one of the key factors in perceptual grouping . This is known as the Law of Symmetry . The role of symmetry in grouping and figure/ground organization has been confirmed in many studies. For instance, detection of reflectional symmetry

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2964-417: The symmetrical nature, often including asymmetrical balance, of social interactions in a variety of contexts. These include assessments of reciprocity , empathy , sympathy , apology , dialogue , respect, justice , and revenge . Reflective equilibrium is the balance that may be attained through deliberative mutual adjustment among general principles and specific judgments . Symmetrical interactions send

3021-814: The terminology: Some authors used letters for connectives: u . {\displaystyle \operatorname {u.} } for conjunction (German's "und" for "and") and o . {\displaystyle \operatorname {o.} } for disjunction (German's "oder" for "or") in early works by Hilbert (1904); N p {\displaystyle Np} for negation, K p q {\displaystyle Kpq} for conjunction, D p q {\displaystyle Dpq} for alternative denial, A p q {\displaystyle Apq} for disjunction, C p q {\displaystyle Cpq} for implication, E p q {\displaystyle Epq} for biconditional in Łukasiewicz in 1929. Such

3078-448: The two atomic formulas P {\displaystyle P} and Q {\displaystyle Q} , rendering the complex formula P ∨ Q {\displaystyle P\lor Q} . Common connectives include negation , disjunction , conjunction , implication , and equivalence . In standard systems of classical logic , these connectives are interpreted as truth functions , though they receive

3135-521: The use of symmetry in carpet and rug patterns spans a variety of cultures. American Navajo Indians used bold diagonals and rectangular motifs. Many Oriental rugs have intricate reflected centers and borders that translate a pattern. Not surprisingly, rectangular rugs have typically the symmetries of a rectangle —that is, motifs that are reflected across both the horizontal and vertical axes (see Klein four-group § Geometry ). As quilts are made from square blocks (usually 9, 16, or 25 pieces to

3192-508: Was present in extrastriate regions of the occipital cortex but not in the primary visual cortex. The extrastriate regions included V3A, V4, V7, and the lateral occipital complex (LOC). Electrophysiological studies have found a late posterior negativity that originates from the same areas. In general, a large part of the visual system seems to be involved in processing visual symmetry, and these areas involve similar networks to those responsible for detecting and recognising objects. People observe

3249-406: Was probably Alban Berg's Quartet , Op. 3 (1910). Tone rows or pitch class sets which are invariant under retrograde are horizontally symmetrical, under inversion vertically. See also Asymmetric rhythm . The relationship of symmetry to aesthetics is complex. Humans find bilateral symmetry in faces physically attractive; it indicates health and genetic fitness. Opposed to this

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