Associative Operation

An Associative Operation is an binary operation whose operation outcome is independent of the grouping of consecutive occurrences of the operation (Operation Instantiation).

• Context:
  • It satisfies the Associative Law:

[math]\displaystyle{ \forall x,y,z \in S : (x \star y) \star z = x \star (y \star z) }[/math]

where [math]\displaystyle{ \star }[/math] is a binary operation on the set [math]\displaystyle{ S }[/math]

• Example(s):
  • Addition Operation, e.g. [math]\displaystyle{ 5+(2+1) ≡ (5+2)+1 }[/math] or [math]\displaystyle{ +(5,+(2,1)) \equiv +(+(5,2),1) }[/math]
  • Logical Disjunction Operation, e.g. (X∨Y)∨Z ≡ X∨(Y∨Z)
• Counter-Example(s):
  • A Commutative Operation, e.g. 5+2 ≡ 2+5.
  • A Division Operation.
  • A DeMorgan's Law.
• See: Distributive Operation, Associativity Axiom.

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References

2009

• (Wikipedia, 2009) ⇒ http://en.wikipedia.org/wiki/Associativity
  • In mathematics, associativity is a property that a binary operation can have. It means that, within an expression containing two or more of the same associative operators in a row, the order that 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 for instance the equation (5+2)+1=5+(2+1)=8 Even though the parentheses were rearranged (the left side requires adding 5 and 2 first, then adding 1 to the result, whereas the right side requires adding 2 and 1 first, then 5), the value of the expression was not altered. Since this holds true when performing addition on any real numbers, we say that "addition of real numbers is an associative operation."
  • Associativity is not to be confused with commutativity. Commutativity justifies changing the order or sequence of the operands within an expression while associativity does not. For example, (5+2)+1=5+(2+1) is an example of associativity because the parentheses were changed (and consequently the order of operations during evaluation) while the operands 5, 2, and 1 appeared in the exact same order from left to right in the expression.
  • (5+2)+1=(2+5)+1 is not an example of associativity because the operand sequence changed when the 2 and 5 switched places.
  • Associative operations are abundant in mathematics, and in fact most algebraic structures explicitly require their binary operations to be associative. However, many important and interesting operations are non-associative; one common example would be the vector cross product.
• (Wikipedia, 2009) ⇒ http://en.wikipedia.org/wiki/Commutative#Associativity
  • The associative property is closely related to the commutative property. The associative property states that the order in which operations are performed does not affect the final result. In contrast, the commutative property states that the order of the terms does not affect the final result.

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