Associativity

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An Associativity is a mathematical property in which a binary operation, [math]\displaystyle{ \star }[/math] on a set [math]\displaystyle{ S }[/math] satisfies the following equation [math]\displaystyle{ (x \star y) \star z = x \star (y \star z) }[/math] for all [math]\displaystyle{ x, y, z }[/math] in [math]\displaystyle{ S }[/math].

References

2017a

  • (Wikipedia, 2017) ⇒ https://en.wikipedia.org/wiki/Associative_property Retrieved:2017-5-21.
    • In mathematics, the associative property is a property of some binary operations. In propositional logic, associativity is a valid rule of replacement for expressions in logical proofs.

      Within an expression containing two or more occurrences in a row of the same associative operator, the order in which the operations are performed does not matter as long as the sequence of the operands is not changed. That is, rearranging the parentheses in such an expression will not change its value. Consider the following equations: : [math]\displaystyle{ (2 + 3) + 4 = 2 + (3 + 4) = 9 \, }[/math] : [math]\displaystyle{ 2 \times (3 \times 4) = (2 \times 3) \times 4 = 24 . }[/math] Even though the parentheses were rearranged on each line, the values of the expressions were not altered. Since this holds true when performing addition and multiplication on any real numbers, it can be said that "addition and multiplication of real numbers are associative operations".

      Associativity is not to be confused with commutativity, which addresses whether or not the order of two operands changes the result. For example, the order doesn't matter in the multiplication of real numbers, that is, , so we say that the multiplication of real numbers is a commutative operation.

      Associative operations are abundant in mathematics; in fact, many algebraic structures (such as semigroups and categories) explicitly require their binary operations to be associative.

      However, many important and interesting operations are non-associative; some examples include subtraction, exponentiation and the vector cross product. In contrast to the theoretical properties of real numbers, the addition of floating point numbers in computer science is not associative, and the choice of how to associate an expression can have a significant effect on rounding error.

2017b

[math]\displaystyle{ a+(b+c) = (a+b) + c\ \ \text{and}\ \ a(bc) = (ab)c\ }[/math].
A general binary operation [math]\displaystyle{ \star }[/math] is associative (or, which is the same thing, satisfies the law of associativity) if the identity
[math]\displaystyle{ a \star (b \star c) = (a \star b) \star c }[/math]
is valid in the given algebraic system. In a similar manner, associativity of an [math]\displaystyle{ n }[/math]-ary operation [math]\displaystyle{ \omega }[/math] is defined by the identities
[math]\displaystyle{ (x_1 x_2 \ldots x_n)\omega x_{n+1} \ldots x_{2n-1} \omega = x_1 \ldots x_i (x_{i+1} \ldots x_{i+n})\omega x_{i+n+1} \ldots x_{i+2n-1} \omega }[/math]
for all [math]\displaystyle{ i=1,\ldots,n }[/math].
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