General Definition
The notation + is usually reserved for commutative binary operations, i.e. such that x + y = y + x, for all x, y . If such an operation admits an identity element o (such that x + o ( = o + x ) = x for all x), then this element is unique ( o′ = o′ + o = o ). For a given x , if there exists x′ such that x + x′ ( = x′ + x ) = o , then x′ is called an additive inverse of x.
If + is associative (( x + y ) + z = x + ( y + z ) for all x, y, z), then an additive inverse is unique
- x″ = x″ + o = x″ + (x + x′) = (x″ + x) + x′ = o + x′ = x′
We often write x − y as x + (−y).
For example, since addition of real numbers is associative, each real number has a unique additive inverse.
Read more about this topic: Additive Inverse
Famous quotes containing the words general and/or definition:
“We have wasted our spirit in the regions of the abstract and general just as the monks let it wither in the world of prayer and contemplation.”
—Alexander Herzen (1812–1870)
“Mothers often are too easily intimidated by their children’s negative reactions...When the child cries or is unhappy, the mother reads this as meaning that she is a failure. This is why it is so important for a mother to know...that the process of growing up involves by definition things that her child is not going to like. Her job is not to create a bed of roses, but to help him learn how to pick his way through the thorns.”
—Elaine Heffner (20th century)