Associativity

Associative property is a property of mathematical operations (like addition and multiplication). It means that if you have more than one of the same associative operator (like +) in a row, the order of operations does not matter.

For example, if you have [math]\displaystyle{ 2+5+10\ }[/math], there are two plus signs (+) in a row. This means we can add it in either this order:

[math]\displaystyle{ (2+5)+10=(7)+10=17\ }[/math]

Or this order:

[math]\displaystyle{ 2+(5+10)=2+(15)=17\ }[/math]

The answer comes out the same both ways because addition is associative. In other words, associativity means:

[math]\displaystyle{ (2+5)+10=2+(5+10)\ }[/math]

Not all operations are associative. Subtraction is not associative, which means:

[math]\displaystyle{ (10-5)-2\ne10-(5-2) }[/math]

This is true because:

[math]\displaystyle{ (10-5)-2=(5)-2=3\ }[/math]

[math]\displaystyle{ 10-(5-2)=10-(3)=7\ }[/math]

And:

[math]\displaystyle{ 7\ne3 }[/math]

Also, associativity is different from commutativity, which lets you move the numbers around.

Associativity Media

A binary operation ∗ on the set S is associative when this diagram commutes. That is, when the two paths from $S \times S \times S$ to $S$ compose to the same function from $S \times S \times S$ to $S$ .
In the absence of the associative property, five factors $a$ , $b$ , $c$ , $d$ , $e$ result in a Tamari lattice of order four, possibly different products.
The addition of real numbers is associative.