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Not all operators or functions are anti-commutative; addition for example is commutative, not anti-commutative. Exponentiation is not commutative (2^3 != 3^2) a

Not all operators or functions are anti-commutative; addition for example is commutative, not anti-commutative. Exponentiation is not commutative (2^3 != 3^2) and it is not anti-commutative because it has no identity (there are exponentiation equivalencies (sometimes labelled identities) but there is no symmetric identity like 0 for addition and subtraction or 1 for multiplication or division). An operator or function is called anti-commutative when it fits the necessary conditions, not the converse. You can't just label something as anti-commutative and then point out that is doesn't fit the conditions; that's putting the cart in front of the donkey.

taejo 9y ago
You wrote: > it also applies to any operator or function. I understand what you mean (that anticommutativity is a property that any binary operator might have) but it did sound like you meant all operators are anticommutative.
- thaumasiotes 9y ago
  He's wrong about anticommutativity, too, though. An operation ∘ is anticommutative if a∘b = -(b∘a). It happens that this is true of subtraction, but that has nothing to do with 0 being the "subtractive identity" or with the - in front of "-(b-a)" visually resembling the sign for subtraction. Vector cross product is anticommutative too, because a×b = -(b×a), but 0 is not an identity for vector cross product and negating a vector is not the same operation as taking the cross product of two vectors. The 0 and the negation come from the operation of adding vectors, not cross-multiplying them.
thaumasiotes 9y ago
> there is no symmetric identity like 0 for addition and subtraction or 1 for multiplication or division 0 is an identity for addition, but not for subtraction. 1 is an identity for multiplication, but not for division. They are right identities for these operations, but 1 is also a right identity for exponentiation.

taejo 9y ago

You wrote: > it also applies to any operator or function. I understand what you mean (that anticommutativity is a property that any binary operator might have) but it did sound like you meant all operators are anticommutative.

thaumasiotes 9y ago
He's wrong about anticommutativity, too, though. An operation ∘ is anticommutative if a∘b = -(b∘a). It happens that this is true of subtraction, but that has nothing to do with 0 being the "subtractive identity" or with the - in front of "-(b-a)" visually resembling the sign for subtraction. Vector cross product is anticommutative too, because a×b = -(b×a), but 0 is not an identity for vector cross product and negating a vector is not the same operation as taking the cross product of two vectors. The 0 and the negation come from the operation of adding vectors, not cross-multiplying them.

thaumasiotes 9y ago

> there is no symmetric identity like 0 for addition and subtraction or 1 for multiplication or division 0 is an identity for addition, but not for subtraction. 1 is an identity for multiplication, but not for division. They are right identities for these operations, but 1 is also a right identity for exponentiation.