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For example: : 4 • (2 + 3) = (4 • 2) + (4 • 3).+ 1 In the left-hand side of the above equation, the 4 multiplies the sum of 2 and 3; on the right-hand side, it multiplies the 2 and the 3 individually, with the results added afterwards. Because these give the same final answer (20), we say that multiplication by 4 ''distributes'' over addition of 2 and 3. Since we could have put any Real Number s in place of 4, 2, and 3 above, and still have obtained a true equation, we say that Multiplication of real numbers ''distributes'' over Addition of real numbers. DEFINITION Given a Set ''S'' and two Binary Operation s • and + on ''S'', we say that
::''x'' • (''y'' + ''z'') = (''x'' • ''y'') + (''x'' • ''z'');
::(''y'' + ''z'') • ''x'' = (''y'' • ''x'') + (''z'' • ''x'');
Notice that when • is Commutative , then the three above conditions are Logically Equivalent . EXAMPLES # Multiplication of Number s is distributive over addition of numbers, for a broad class of different kinds of numbers ranging from Natural Number s to Complex Number s and Cardinal Number s. # Multiplication of Ordinal Number s, in contrast, is only left-distributive, not right-distributive. # Matrix Multiplication is distributive over Matrix Addition , even though it's not commutative. # The Union of Set s is distributive over Intersection , and intersection is distributive over union. Also, intersection is distributive over the Symmetric Difference . # Logical Disjunction ("or") is distributive over Logical Conjunction ("and"), and conjunction is distributive over disjunction. Also, conjunction is distributive over Exclusive Disjunction ("xor"). # For Real Number s (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 Integer s, 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 is lost around the limits of 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 Ring s and Distributive Lattice s.
Most kinds of numbers (example 1) and matrices (example 3) form rings. A Lattice is another kind of Algebraic Structure with two binary operations, ^ and v. If either of these operations (say ^) distributes over the other (v), then v must also distribute over ^, and the lattice is called distributive. See also the article on Distributivity (order Theory) . Examples 4 and 5 are Boolean Algebra s, 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 Rig s, certain generalisations of rings. Those numbers in example 1 that don't form rings at least form rigs. Near-rig s are a further generalisation 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 Implication Operator of Heyting Algebra s. Details of 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 Intervals . In . EXTERNAL LINKS
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