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In mathematics, the associative property [1] is a property of some binary operations that means that rearranging the parentheses in an expression will not change the result. In propositional logic , associativity is a valid rule of replacement for expressions in logical proofs .
The Egyptians used the commutative property of multiplication to simplify computing products. [7] [8] Euclid is known to have assumed the commutative property of multiplication in his book Elements. [9] Formal uses of the commutative property arose in the late 18th and early 19th centuries, when mathematicians began to work on a theory of ...
The base case b = 0 follows immediately from the identity element property (0 is an additive identity), which has been proved above: a + 0 = a = 0 + a. Next we will prove the base case b = 1, that 1 commutes with everything, i.e. for all natural numbers a, we have a + 1 = 1 + a.
The natural sum is associative and commutative. It is always greater or equal to the usual sum, but it may be strictly greater. For example, the natural sum of ω and 1 is ω + 1 (the usual sum), but this is also the natural sum of 1 and ω. The natural product is associative and commutative and distributes over the natural sum.
The algebra of sets is the set-theoretic analogue of the algebra of numbers. Just as arithmetic addition and multiplication are associative and commutative, so are set union and intersection; just as the arithmetic relation "less than or equal" is reflexive, antisymmetric and transitive, so is the set relation of "subset".
Calculators generally perform operations with the same precedence from left to right, [1] but some programming languages and calculators adopt different conventions. For example, multiplication is granted a higher precedence than addition, and it has been this way since the introduction of modern algebraic notation.
Unfortunately, it is not generally possible to determine whether or not an operation is associative simply by glancing at its Cayley table, as it is with commutativity. This is because associativity depends on a 3 term equation, ( a b ) c = a ( b c ) {\displaystyle (ab)c=a(bc)} , while the Cayley table shows 2-term products.
In mathematics, an associative algebra A over a commutative ring (often a field) K is a ring A together with a ring homomorphism from K into the center of A.This is thus an algebraic structure with an addition, a multiplication, and a scalar multiplication (the multiplication by the image of the ring homomorphism of an element of K).