Closure.md (1174B)
1 # Closure 2 3 Khan 4 5 **Definition:** Closure means that performing some arbitrary operation (pick one, but not necessarily all) on any member of a set will result in another element of a set. 6 7 In the context of subspaces, we have closure under scalar multiplication and vector addition because these operations on any element of the [LinearSubspace](LinearSubspace.md) set results in another element of the set (by definition). 8 9 ## Discrete Math 10 11 **Definition:** A closure is when we have a property such that the relation contains the original set of the relation and has the minimum number of extra components to be closed with respect to the property. 12 13 Closure under addition means we have the minimum number of elements in a relation such that the domain and codomain are included in the first and second components of ordered pairs, and all other elements necessary for addition to result in another element of the relation are included. 14 15 An important note is that there are often many relations for which the relation has the property and contains the codomain and domain, but we are only interested in the smallest one, the one that has the fewest other elements in it.