In mathematics , the distributive property of binary operations is a generalization of the distributive law , which asserts that the equality x ⋅ ( y + z ) = x ⋅ y + x ⋅ z {\displaystyle x\cdot (y+z)=x\cdot y+x\cdot z} is always true in elementary algebra . For example, in elementary arithmetic , one has 2 ⋅ ( 1 + 3 ) = ( 2 ⋅ 1 ) + ( 2 ⋅ 3 ) . {\displaystyle 2\cdot (1+3)=(2\cdot 1)+(2\cdot 3).} Therefore, one would say that multiplication distributes over addition .
50-410: Distributive may refer to: Distributive property , in algebra, logic and mathematics Distributive pronoun and distributive adjective (determiner), in linguistics Distributive case , in linguistics Distributive numeral , in linguistics Topics referred to by the same term [REDACTED] This disambiguation page lists articles associated with
100-410: A ÷ c ± b ÷ c . {\displaystyle (a\pm b)\div c=a\div c\pm b\div c.} In this case, left-distributivity does not apply: a ÷ ( b ± c ) ≠ a ÷ b ± a ÷ c {\displaystyle a\div (b\pm c)\neq a\div b\pm a\div c} The distributive laws are among
150-518: A ∨ b ) ⇒ c ≡ ( a ⇒ c ) ∧ ( b ⇒ c ) {\displaystyle (a\lor b)\Rightarrow c\equiv (a\Rightarrow c)\land (b\Rightarrow c)} ( a ∧ b ) ⇒ c ≡ ( a ⇒ c ) ∨ ( b ⇒ c ) . {\displaystyle (a\land b)\Rightarrow c\equiv (a\Rightarrow c)\lor (b\Rightarrow c).} These two tautologies are
200-414: A ⋅ c ± b ⋅ c (right-distributive) . {\displaystyle (a\pm b)\cdot c=a\cdot c\pm b\cdot c\qquad {\text{ (right-distributive) }}.} In either case, the distributive property can be described in words as: To multiply a sum (or difference ) by a factor, each summand (or minuend and subtrahend ) is multiplied by this factor and
250-589: A category C , {\displaystyle C,} a distributive law S . S ′ → S ′ . S {\displaystyle S.S^{\prime }\to S^{\prime }.S} is a natural transformation λ : S . S ′ → S ′ . S {\displaystyle \lambda :S.S^{\prime }\to S^{\prime }.S} such that ( S ′ , λ ) {\displaystyle \left(S^{\prime },\lambda \right)}
300-641: A set S is called commutative if x ∗ y = y ∗ x for all x , y ∈ S . {\displaystyle x*y=y*x\qquad {\mbox{for all }}x,y\in S.} In other words, an operation is commutative if every two elements commute. An operation that does not satisfy the above property is called noncommutative . One says that x commutes with y or that x and y commute under ∗ {\displaystyle *} if x ∗ y = y ∗ x . {\displaystyle x*y=y*x.} That is,
350-406: A unary operation ). In the context of a near-ring , which removes the commutativity of the additively written group and assumes only one-sided distributivity, one can speak of (two-sided) distributive elements but also of antidistributive elements . The latter reverse the order of (the non-commutative) addition; assuming a left-nearring (i.e. one which all elements distribute when multiplied on
400-1033: A vector space to itself (see below for the Matrix representation). Matrix multiplication of square matrices is almost always noncommutative, for example: [ 0 2 0 1 ] = [ 1 1 0 1 ] [ 0 1 0 1 ] ≠ [ 0 1 0 1 ] [ 1 1 0 1 ] = [ 0 1 0 1 ] {\displaystyle {\begin{bmatrix}0&2\\0&1\end{bmatrix}}={\begin{bmatrix}1&1\\0&1\end{bmatrix}}{\begin{bmatrix}0&1\\0&1\end{bmatrix}}\neq {\begin{bmatrix}0&1\\0&1\end{bmatrix}}{\begin{bmatrix}1&1\\0&1\end{bmatrix}}={\begin{bmatrix}0&1\\0&1\end{bmatrix}}} The vector product (or cross product ) of two vectors in three dimensions
450-461: A direct consequence of the duality in De Morgan's laws . Commutative In mathematics , a binary operation is commutative if changing the order of the operands does not change the result. It is a fundamental property of many binary operations, and many mathematical proofs depend on it. Perhaps most familiar as a property of arithmetic, e.g. "3 + 4 = 4 + 3" or "2 × 5 = 5 × 2" ,
500-411: A specific pair of elements may commute even if the operation is (strictly) noncommutative. Division is noncommutative, since 1 ÷ 2 ≠ 2 ÷ 1 {\displaystyle 1\div 2\neq 2\div 1} . Subtraction is noncommutative, since 0 − 1 ≠ 1 − 0 {\displaystyle 0-1\neq 1-0} . However it
550-443: A sum is multiplied by a sum, multiply each summand of a sum with each summand of the other sum (keeping track of signs) then add up all of the resulting products. In the following examples, the use of the distributive law on the set of real numbers R {\displaystyle \mathbb {R} } is illustrated. When multiplication is mentioned in elementary mathematics, it usually refers to this kind of multiplication. From
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#1732780275225600-474: A theory of functions. Today the commutative property is a well-known and basic property used in most branches of mathematics. The first recorded use of the term commutative was in a memoir by François Servois in 1814, which used the word commutatives when describing functions that have what is now called the commutative property. Commutative is the feminine form of the French adjective commutatif , which
650-471: Is anti-commutative ; i.e., b × a = −( a × b ). Records of the implicit use of the commutative property go back to ancient times. The Egyptians used the commutative property of multiplication to simplify computing products . Euclid is known to have assumed the commutative property of multiplication in his book Elements . Formal uses of the commutative property arose in the late 18th and early 19th centuries, when mathematicians began to work on
700-416: Is commutative , the three conditions above are logically equivalent . The operators used for examples in this section are those of the usual addition + {\displaystyle \,+\,} and multiplication ⋅ . {\displaystyle \,\cdot .\,} If the operation denoted ⋅ {\displaystyle \cdot } is not commutative, there
750-499: Is a lax map of monads S → S {\displaystyle S\to S} and ( S , λ ) {\displaystyle (S,\lambda )} is a colax map of monads S ′ → S ′ . {\displaystyle S^{\prime }\to S^{\prime }.} This is exactly the data needed to define a monad structure on S ′ . S {\displaystyle S^{\prime }.S} :
800-451: Is a metalogical symbol representing "can be replaced in a proof with". Commutativity is a property of some logical connectives of truth functional propositional logic . The following logical equivalences demonstrate that commutativity is a property of particular connectives. The following are truth-functional tautologies . In group and set theory , many algebraic structures are called commutative when certain operands satisfy
850-6753: Is a metalogical symbol representing "can be replaced in a proof with" or "is logically equivalent to". Distributivity is a property of some logical connectives of truth-functional propositional logic . The following logical equivalences demonstrate that distributivity is a property of particular connectives. The following are truth-functional tautologies . ( P ∧ ( Q ∨ R ) ) ⇔ ( ( P ∧ Q ) ∨ ( P ∧ R ) ) Distribution of conjunction over disjunction ( P ∨ ( Q ∧ R ) ) ⇔ ( ( P ∨ Q ) ∧ ( P ∨ R ) ) Distribution of disjunction over conjunction ( P ∧ ( Q ∧ R ) ) ⇔ ( ( P ∧ Q ) ∧ ( P ∧ R ) ) Distribution of conjunction over conjunction ( P ∨ ( Q ∨ R ) ) ⇔ ( ( P ∨ Q ) ∨ ( P ∨ R ) ) Distribution of disjunction over disjunction ( P → ( Q → R ) ) ⇔ ( ( P → Q ) → ( P → R ) ) Distribution of implication ( P → ( Q ↔ R ) ) ⇔ ( ( P → Q ) ↔ ( P → R ) ) Distribution of implication over equivalence ( P → ( Q ∧ R ) ) ⇔ ( ( P → Q ) ∧ ( P → R ) ) Distribution of implication over conjunction ( P ∨ ( Q ↔ R ) ) ⇔ ( ( P ∨ Q ) ↔ ( P ∨ R ) ) Distribution of disjunction over equivalence {\displaystyle {\begin{alignedat}{13}&(P&&\;\land &&(Q\lor R))&&\;\Leftrightarrow \;&&((P\land Q)&&\;\lor (P\land R))&&\quad {\text{ Distribution of }}&&{\text{ conjunction }}&&{\text{ over }}&&{\text{ disjunction }}\\&(P&&\;\lor &&(Q\land R))&&\;\Leftrightarrow \;&&((P\lor Q)&&\;\land (P\lor R))&&\quad {\text{ Distribution of }}&&{\text{ disjunction }}&&{\text{ over }}&&{\text{ conjunction }}\\&(P&&\;\land &&(Q\land R))&&\;\Leftrightarrow \;&&((P\land Q)&&\;\land (P\land R))&&\quad {\text{ Distribution of }}&&{\text{ conjunction }}&&{\text{ over }}&&{\text{ conjunction }}\\&(P&&\;\lor &&(Q\lor R))&&\;\Leftrightarrow \;&&((P\lor Q)&&\;\lor (P\lor R))&&\quad {\text{ Distribution of }}&&{\text{ disjunction }}&&{\text{ over }}&&{\text{ disjunction }}\\&(P&&\to &&(Q\to R))&&\;\Leftrightarrow \;&&((P\to Q)&&\to (P\to R))&&\quad {\text{ Distribution of }}&&{\text{ implication }}&&{\text{ }}&&{\text{ }}\\&(P&&\to &&(Q\leftrightarrow R))&&\;\Leftrightarrow \;&&((P\to Q)&&\leftrightarrow (P\to R))&&\quad {\text{ Distribution of }}&&{\text{ implication }}&&{\text{ over }}&&{\text{ equivalence }}\\&(P&&\to &&(Q\land R))&&\;\Leftrightarrow \;&&((P\to Q)&&\;\land (P\to R))&&\quad {\text{ Distribution of }}&&{\text{ implication }}&&{\text{ over }}&&{\text{ conjunction }}\\&(P&&\;\lor &&(Q\leftrightarrow R))&&\;\Leftrightarrow \;&&((P\lor Q)&&\leftrightarrow (P\lor R))&&\quad {\text{ Distribution of }}&&{\text{ disjunction }}&&{\text{ over }}&&{\text{ equivalence }}\\\end{alignedat}}} ( ( P ∧ Q ) ∨ ( R ∧ S ) ) ⇔ ( ( ( P ∨ R ) ∧ ( P ∨ S ) ) ∧ ( ( Q ∨ R ) ∧ ( Q ∨ S ) ) ) ( ( P ∨ Q ) ∧ ( R ∨ S ) ) ⇔ ( ( ( P ∧ R ) ∨ ( P ∧ S ) ) ∨ ( ( Q ∧ R ) ∨ ( Q ∧ S ) ) ) {\displaystyle {\begin{alignedat}{13}&((P\land Q)&&\;\lor (R\land S))&&\;\Leftrightarrow \;&&(((P\lor R)\land (P\lor S))&&\;\land ((Q\lor R)\land (Q\lor S)))&&\\&((P\lor Q)&&\;\land (R\lor S))&&\;\Leftrightarrow \;&&(((P\land R)\lor (P\land S))&&\;\lor ((Q\land R)\lor (Q\land S)))&&\\\end{alignedat}}} In approximate arithmetic, such as floating-point arithmetic ,
900-423: Is a distinction between left-distributivity and right-distributivity: a ⋅ ( b ± c ) = a ⋅ b ± a ⋅ c (left-distributive) {\displaystyle a\cdot \left(b\pm c\right)=a\cdot b\pm a\cdot c\qquad {\text{ (left-distributive) }}} ( a ± b ) ⋅ c =
950-460: Is a semiring with additive inverses. A lattice is another kind of algebraic structure with two binary operations, ∧ and ∨ . {\displaystyle \,\land {\text{ and }}\lor .} If either of these operations distributes over the other (say ∧ {\displaystyle \,\land \,} distributes over ∨ {\displaystyle \,\lor } ), then
1000-478: Is always associative but not always commutative. Some forms of symmetry can be directly linked to commutativity. When a commutative operation is written as a binary function z = f ( x , y ) , {\displaystyle z=f(x,y),} then this function is called a symmetric function , and its graph in three-dimensional space is symmetric across the plane y = x {\displaystyle y=x} . For example, if
1050-417: Is classified more precisely as anti-commutative , since 0 − 1 = − ( 1 − 0 ) {\displaystyle 0-1=-(1-0)} . Exponentiation is noncommutative, since 2 3 ≠ 3 2 {\displaystyle 2^{3}\neq 3^{2}} . This property leads to two different "inverse" operations of exponentiation (namely,
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#17327802752251100-1222: Is derived from the French noun commutation and the French verb commuter , meaning "to exchange" or "to switch", a cognate of to commute . The term then appeared in English in 1838. in Duncan Gregory 's article entitled "On the real nature of symbolical algebra" published in 1840 in the Transactions of the Royal Society of Edinburgh . In truth-functional propositional logic, commutation , or commutativity refer to two valid rules of replacement . The rules allow one to transpose propositional variables within logical expressions in logical proofs . The rules are: ( P ∨ Q ) ⇔ ( Q ∨ P ) {\displaystyle (P\lor Q)\Leftrightarrow (Q\lor P)} and ( P ∧ Q ) ⇔ ( Q ∧ P ) {\displaystyle (P\land Q)\Leftrightarrow (Q\land P)} where " ⇔ {\displaystyle \Leftrightarrow } "
1150-541: Is part of the definition of most algebraic structures that have two operations called addition and multiplication, such as complex numbers , polynomials , matrices , rings , and fields . It is also encountered in Boolean algebra and mathematical logic , where each of the logical and (denoted ∧ {\displaystyle \,\land \,} ) and the logical or (denoted ∨ {\displaystyle \,\lor \,} ) distributes over
1200-476: Is responsible for different distributive laws in the Boolean algebra. Similar structures without distributive laws are near-rings and near-fields instead of rings and division rings . The operations are usually defined to be distributive on the right but not on the left. In several mathematical areas, generalized distributivity laws are considered. This may involve the weakening of the above conditions or
1250-782: Is the function f ( x , y ) = x + y 2 , {\displaystyle f(x,y)={\frac {x+y}{2}},} which is clearly commutative (interchanging x and y does not affect the result), but it is not associative (since, for example, f ( − 4 , f ( 0 , + 4 ) ) = − 1 {\displaystyle f(-4,f(0,+4))=-1} but f ( f ( − 4 , 0 ) , + 4 ) = + 1 {\displaystyle f(f(-4,0),+4)=+1} ). More such examples may be found in commutative non-associative magmas . Furthermore, associativity does not imply commutativity either – for example multiplication of quaternions or of matrices
1300-458: Is the notion of sub-distributivity as explained in the article on interval arithmetic . In category theory , if ( S , μ , ν ) {\displaystyle (S,\mu ,\nu )} and ( S ′ , μ ′ , ν ′ ) {\displaystyle \left(S^{\prime },\mu ^{\prime },\nu ^{\prime }\right)} are monads on
1350-419: The n th-root operation and the logarithm operation), whereas multiplication only has one inverse operation. Some truth functions are noncommutative, since the truth tables for the functions are different when one changes the order of the operands. For example, the truth tables for (A ⇒ B) = (¬A ∨ B) and (B ⇒ A) = (A ∨ ¬B) are Function composition of linear functions from the real numbers to
1400-680: The uncertainty principle of Heisenberg , if the two operators representing a pair of variables do not commute, then that pair of variables are mutually complementary , which means they cannot be simultaneously measured or known precisely. For example, the position and the linear momentum in the x {\displaystyle x} -direction of a particle are represented by the operators x {\displaystyle x} and − i ℏ ∂ ∂ x {\displaystyle -i\hbar {\frac {\partial }{\partial x}}} , respectively (where ℏ {\displaystyle \hbar }
1450-474: The area of information theory . The ubiquitous identity that relates inverses to the binary operation in any group , namely ( x y ) − 1 = y − 1 x − 1 , {\displaystyle (xy)^{-1}=y^{-1}x^{-1},} which is taken as an axiom in the more general context of a semigroup with involution , has sometimes been called an antidistributive property (of inversion as
1500-434: The axioms for rings (like the ring of integers ) and fields (like the field of rational numbers ). Here multiplication is distributive over addition, but addition is not distributive over multiplication. Examples of structures with two operations that are each distributive over the other are Boolean algebras such as the algebra of sets or the switching algebra . Multiplying sums can be put into words as follows: When
1550-446: The commutative property does not hold for matrix multiplication, the second law does not follow from the first law. In this case, they are two different laws. In standard truth-functional propositional logic, distribution in logical proofs uses two valid rules of replacement to expand individual occurrences of certain logical connectives , within some formula , into separate applications of those connectives across subformulas of
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1600-422: The commutative property. In higher branches of mathematics, such as analysis and linear algebra the commutativity of well-known operations (such as addition and multiplication on real and complex numbers) is often used (or implicitly assumed) in proofs. The associative property is closely related to the commutative property. The associative property of an expression containing two or more occurrences of
1650-503: The distributive property of multiplication (and division) over addition may fail because of the limitations of arithmetic precision . For example, the identity 1 / 3 + 1 / 3 + 1 / 3 = ( 1 + 1 + 1 ) / 3 {\displaystyle 1/3+1/3+1/3=(1+1+1)/3} fails in decimal arithmetic , regardless of the number of significant digits . Methods such as banker's rounding may help in some cases, as may increasing
1700-865: The effect of their compositions x d d x {\textstyle x{\frac {d}{dx}}} and d d x x {\textstyle {\frac {d}{dx}}x} (also called products of operators) on a one-dimensional wave function ψ ( x ) {\displaystyle \psi (x)} : x ⋅ d d x ψ = x ⋅ ψ ′ ≠ ψ + x ⋅ ψ ′ = d d x ( x ⋅ ψ ) {\displaystyle x\cdot {\mathrm {d} \over \mathrm {d} x}\psi =x\cdot \psi '\ \neq \ \psi +x\cdot \psi '={\mathrm {d} \over \mathrm {d} x}\left(x\cdot \psi \right)} According to
1750-413: The extension to infinitary operations. Especially in order theory one finds numerous important variants of distributivity, some of which include infinitary operations, such as the infinite distributive law ; others being defined in the presence of only one binary operation, such as the according definitions and their relations are given in the article distributivity (order theory) . This also includes
1800-826: The function f is defined as f ( x , y ) = x + y {\displaystyle f(x,y)=x+y} then f {\displaystyle f} is a symmetric function. For relations, a symmetric relation is analogous to a commutative operation, in that if a relation R is symmetric, then a R b ⇔ b R a {\displaystyle aRb\Leftrightarrow bRa} . In quantum mechanics as formulated by Schrödinger , physical variables are represented by linear operators such as x {\displaystyle x} (meaning multiply by x {\displaystyle x} ), and d d x {\textstyle {\frac {d}{dx}}} . These two operators do not commute as may be seen by considering
1850-730: The given formula. The rules are ( P ∧ ( Q ∨ R ) ) ⇔ ( ( P ∧ Q ) ∨ ( P ∧ R ) ) and ( P ∨ ( Q ∧ R ) ) ⇔ ( ( P ∨ Q ) ∧ ( P ∨ R ) ) {\displaystyle (P\land (Q\lor R))\Leftrightarrow ((P\land Q)\lor (P\land R))\qquad {\text{ and }}\qquad (P\lor (Q\land R))\Leftrightarrow ((P\lor Q)\land (P\lor R))} where " ⇔ {\displaystyle \Leftrightarrow } ", also written ≡ , {\displaystyle \,\equiv ,\,}
1900-493: The left), then an antidistributive element a {\displaystyle a} reverses the order of addition when multiplied to the right: ( x + y ) a = y a + x a . {\displaystyle (x+y)a=ya+xa.} In the study of propositional logic and Boolean algebra , the term antidistributive law is sometimes used to denote the interchange between conjunction and disjunction when implication factors over them: (
1950-512: The multiplication map is S ′ μ . μ ′ S 2 . S ′ λ S {\displaystyle S^{\prime }\mu .\mu ^{\prime }S^{2}.S^{\prime }\lambda S} and the unit map is η ′ S . η . {\displaystyle \eta ^{\prime }S.\eta .} See: distributive law between monads . A generalized distributive law has also been proposed in
2000-455: The notion of a completely distributive lattice . In the presence of an ordering relation, one can also weaken the above equalities by replacing = {\displaystyle \,=\,} by either ≤ {\displaystyle \,\leq \,} or ≥ . {\displaystyle \,\geq .} Naturally, this will lead to meaningful concepts only in some situations. An application of this principle
2050-708: The other. Given a set S {\displaystyle S} and two binary operators ∗ {\displaystyle \,*\,} and + {\displaystyle \,+\,} on S , {\displaystyle S,} x ∗ ( y + z ) = ( x ∗ y ) + ( x ∗ z ) ; {\displaystyle x*(y+z)=(x*y)+(x*z);} ( y + z ) ∗ x = ( y ∗ x ) + ( z ∗ x ) ; {\displaystyle (y+z)*x=(y*x)+(z*x);} When ∗ {\displaystyle \,*\,}
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2100-1116: The point of view of algebra, the real numbers form a field , which ensures the validity of the distributive law. The distributive law is valid for matrix multiplication . More precisely, ( A + B ) ⋅ C = A ⋅ C + B ⋅ C {\displaystyle (A+B)\cdot C=A\cdot C+B\cdot C} for all l × m {\displaystyle l\times m} -matrices A , B {\displaystyle A,B} and m × n {\displaystyle m\times n} -matrices C , {\displaystyle C,} as well as A ⋅ ( B + C ) = A ⋅ B + A ⋅ C {\displaystyle A\cdot (B+C)=A\cdot B+A\cdot C} for all l × m {\displaystyle l\times m} -matrices A {\displaystyle A} and m × n {\displaystyle m\times n} -matrices B , C . {\displaystyle B,C.} Because
2150-544: The precision used, but ultimately some calculation errors are inevitable. Distributivity is most commonly found in semirings , notably the particular cases of rings and distributive lattices . A semiring has two binary operations, commonly denoted + {\displaystyle \,+\,} and ∗ , {\displaystyle \,*,} and requires that ∗ {\displaystyle \,*\,} must distribute over + . {\displaystyle \,+.} A ring
2200-455: The property can also be used in more advanced settings. The name is needed because there are operations, such as division and subtraction , that do not have it (for example, "3 − 5 ≠ 5 − 3" ); such operations are not commutative, and so are referred to as noncommutative operations . The idea that simple operations, such as the multiplication and addition of numbers, are commutative was for many years implicitly assumed. Thus, this property
2250-828: The real numbers is almost always noncommutative. For example, let f ( x ) = 2 x + 1 {\displaystyle f(x)=2x+1} and g ( x ) = 3 x + 7 {\displaystyle g(x)=3x+7} . Then ( f ∘ g ) ( x ) = f ( g ( x ) ) = 2 ( 3 x + 7 ) + 1 = 6 x + 15 {\displaystyle (f\circ g)(x)=f(g(x))=2(3x+7)+1=6x+15} and ( g ∘ f ) ( x ) = g ( f ( x ) ) = 3 ( 2 x + 1 ) + 7 = 6 x + 10 {\displaystyle (g\circ f)(x)=g(f(x))=3(2x+1)+7=6x+10} This also applies more generally for linear and affine transformations from
2300-423: The resulting products are added (or subtracted). If the operation outside the parentheses (in this case, the multiplication) is commutative, then left-distributivity implies right-distributivity and vice versa, and one talks simply of distributivity . One example of an operation that is "only" right-distributive is division, which is not commutative: ( a ± b ) ÷ c =
2350-439: The reverse also holds ( ∨ {\displaystyle \,\lor \,} distributes over ∧ {\displaystyle \,\land \,} ), and the lattice is called distributive. See also Distributivity (order theory) . A Boolean algebra can be interpreted either as a special kind of ring (a Boolean ring ) or a special kind of distributive lattice (a Boolean lattice ). Each interpretation
2400-401: The same operator states that the order operations are performed in does not affect the final result, as long as the order of terms does not change. In contrast, the commutative property states that the order of the terms does not affect the final result. Most commutative operations encountered in practice are also associative. However, commutativity does not imply associativity. A counterexample
2450-494: The title Distributive . If an internal link led you here, you may wish to change the link to point directly to the intended article. Retrieved from " https://en.wikipedia.org/w/index.php?title=Distributive&oldid=947936782 " Category : Disambiguation pages Hidden categories: Short description is different from Wikidata All article disambiguation pages All disambiguation pages Distributive property This basic property of numbers
2500-434: Was not named until the 19th century, when mathematics started to become formalized. A similar property exists for binary relations ; a binary relation is said to be symmetric if the relation applies regardless of the order of its operands; for example, equality is symmetric as two equal mathematical objects are equal regardless of their order. A binary operation ∗ {\displaystyle *} on
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