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The COMMUTATIVE property states that the order of the arguments of an operation does not matter. In symbolic terms, for elements a and b and for the operation ~, a ~ b = b ~ a

The ASSOCIATIVE property states that the order in which the operation is carried out does not matter. Symbolically, for elements a, b and c, (a ~ b) ~ c = a ~ (b ~ c) and so, without ambiguity, either can be written as a ~ b ~ c.

The DISTRIBUTIVE property is a property of two operations, for example, of multiplication over addition. It is not the property of a single operation. For operations ~ and # and elements a, b and c, symbolically, this means that a ~ (b # c) = a ~ b # a ~ c.

The existence of an IDENTITY is a property of the set over which the operation ~ is defined; not a property of operation itself. Symbolically, if the identity exists, it is a unique element, denoted by i, such that a ~ i = a = i ~ a for all a in the set.

For example, you can define addition on all positive integers which will have the commutative and associative properties but the identity (zero) and additive inverses (negative numbers) are undefined as far as the set is concerned.

I have deliberately chosen ~ and # to represent the operations rather than addition or multiplication because there are circumstances in which these properties do not apply to multiplication (for example for matrices), and there are many other operations that they can apply to.

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Q: What is the distributive property the commutative property the associate property and the identity property?
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