In abstract algebra and logic, distributivity is a property of binary operations that generalizes the distributive law from elementary algebra. In propositional logic, distribution refers to two valid rules of replacement. The rules allow one to reformulate conjunctions and disjunctions within logical proofs.
For example, in arithmetic:
- 2 (1 + 3) = (2 1) + (2 3).
In the left-hand side of the first equation, the 2 multiplies the sum of 1 and 3; on the right-hand side, it multiplies the 1 and the 3 individually, with the products added afterwards. Because these give the same final answer (8), we say that multiplication by 2 distributes over addition of 1 and 3. Since we could have put any real numbers in place of 2, 1, and 3 above, and still have obtained a true equation, we say that multiplication of real numbers distributes over addition of real numbers.
Given a set S and two binary operators and + on S, we say that the operation
- is left-distributive over + if, given any elements x, y, and z of S,
x (y + z) = (x y) + (x z);
- is right-distributive over + if, given any elements x, y, and z of S:
- (y + z) x = (y x) + (z x);
- is distributive over + if it is left- and right-distributive.
Notice that when is commutative, then the three above conditions are logically equivalent.
Rule of replacement
In standard truth-functional propositional logic, distribution are two valid rule of replacement. The rules allow one to distribute certain logical connectives within logical expressions in logical proofs. The rules are:
- (P \and (Q \or R)) \Leftrightarrow ((P \and Q) \or (P \and R))
- (P \or (Q \and R)) \Leftrightarrow ((P \or Q) \and (P \or R))
where "\Leftrightarrow" is a metalogical symbol representing "can be replaced in a proof with."
Truth functional connectives
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.
Distribution of conjunction over conjunction
- (P \and (Q \and R)) \leftrightarrow ((P \and Q) \and (P \and R))
Distribution of conjunction over disjunction
- (P \and (Q \or R)) \leftrightarrow ((P \and Q) \or (P \and R))
Distribution of disjunction over conjunction
- (P \or (Q \and R)) \leftrightarrow ((P \or Q) \and (P \or R))
Distribution of disjunction over disjunction
- (P \or (Q \or R)) \leftrightarrow ((P \or Q) \or (P \or R))
Distribution of implication
- (P \to (Q \to R)) \to ((P \to Q) \to (P \to R))
Distribution of implication over equivalence
- P \to (Q \leftrightarrow R) \leftrightarrow ((P \to Q) \leftrightarrow (P \to R))
Distribution of disjunction over equivalence
- (P \or (Q \leftrightarrow R)) \leftrightarrow ((P \or Q) \leftrightarrow (P \or R))
Distribution of negation over equivalence
- \neg (P \leftrightarrow Q) \leftrightarrow (\neg P \leftrightarrow Q)
- ((P \and Q) \or (R \and S)) \leftrightarrow (((P \or R) \and (P \or S)) \and ((Q \or R) \and (Q \or S)))
- ((P \or Q) \and (R \or S)) \leftrightarrow (((P \and R) \or (P \and S)) \or ((Q \and R) \or (Q \and S)))
Self distributive law of implication
- S \to (P \to Q) \to ((S \to P) \to (S \to Q))
- Multiplication of numbers is distributive over addition of numbers, for a broad class of different kinds of numbers ranging from natural numbers to complex numbers and cardinal numbers.
- Multiplication of ordinal numbers, in contrast, is only left-distributive, not right-distributive.
- The cross product is left- and right-distributive over vector addition, though not commutative.
- Matrix multiplication is distributive over matrix addition, though also not commutative.
- The union of sets is distributive over intersection, and intersection is distributive over union.
- Logical disjunction ("or") is distributive over logical conjunction ("and"), and conjunction is distributive over disjunction.
- For real numbers (or for any totally ordered set), the maximum operation is distributive over the minimum operation, and vice versa: max(a,min(b,c)) = min(max(a,b),max(a,c)) and min(a,max(b,c)) = max(min(a,b),min(a,c)).
- For integers, the greatest common divisor is distributive over the least common multiple, and vice versa: gcd(a,lcm(b,c)) = lcm(gcd(a,b),gcd(a,c)) and lcm(a,gcd(b,c)) = gcd(lcm(a,b),LCM(a,c)).
- For real numbers, addition distributes over the maximum operation, and also over the minimum operation: a + max(b,c) = max(a+b,a+c) and a + min(b,c) = min(a+b,a+c).
Distributivity and rounding
In practice, the distributive property of multiplication (and division) over addition may appear to be compromised or lost because of the limitations of arithmetic precision. For example, the identity + + = (1+1+1)/3 appears to fail if the addition is conducted in decimal arithmetic; however, if many significant digits are used, the calculation will result in a closer approximation to the correct results. For example, if the arithmetical calculation takes the form: 0.33333+0.33333+0.33333 = 0.99999 1, this result is a closer approximation than if fewer significant digits had been used. Even when fractional numbers can be represented exactly in arithmetical form, errors will be introduced if those arithmetical values are rounded or truncated. For example, buying two books, each priced at 14.99 before a tax of 17.5%, in two separate transactions will actually save 0.01, over buying them together: 14.99 1.175 = 17.61 to the nearest 0.01, giving a total expenditure of 35.22, but 29.98 1.175 = 35.23. Methods such as banker's rounding may help in some cases, as may increasing the precision used, but ultimately some calculation errors are inevitable.
Distributivity in rings
Distributivity is most commonly found in rings and distributive lattices.
A ring has two binary operations (commonly called "+" and "*"), and one of the requirements of a ring is that * must distribute over +. Most kinds of numbers (example 1) and matrices (example 4) form rings. A lattice is another kind of algebraic structure with two binary operations, and . If either of these operations (say ) distributes over the other ( ), then must also distribute over , and the lattice is called distributive. See also the article on distributivity (order theory).
Examples 4 and 5 are Boolean algebras, which 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 is responsible for different distributive laws in the Boolean algebra. Examples 6 and 7 are distributive lattices which are not Boolean algebras.
Rings and distributive lattices are both special kinds of rigs, certain generalizations of rings. Those numbers in example 1 that don't form rings at least form rigs. Near-rigs are a further generalization of rigs that are left-distributive but not right-distributive; example 2 is a near-rig.
Generalizations of distributivity
In several mathematical areas, generalized distributivity laws are considered. This may involve the weakening of the above conditions or 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 the notion of a completely distributive lattice.
In the presence of an ordering relation, one can also weaken the above equalities by replacing = by either or . Naturally, this will lead to meaningful concepts only in some situations. An application of this principle is the notion of sub-distributivity as explained in the article on interval arithmetic.
In category theory, if (S, , ) and (S', ', ') are monads on a category C, a distributive law S.S' S'.S is a natural transformation : S.S' S'.S such that (S' , ) is a lax map of monads S S and (S, ) is a colax map of monads S' S' . This is exactly the data needed to define a monad structure on S'.S: the multiplication map is S' . 'S .S' S and the unit map is 'S. . See: distributive law between monads.
A generalized distributive law has also been proposed in the area of information theory.
- Ayres, Frank, Schaum's Outline of Modern Abstract Algebra, McGraw-Hill; 1st edition (June 1, 1965). ISBN 0-07-002655-6.
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