 Notes > Models of Computation > Set Representation and Laws
|
Set representation
Some pre-defined special sets are listed below:
- N - Natural numbers
- Z - Integers (both positive and negative including zero)
- R - Real numbers (any number that can be written as a decimal expansion)
- Null (Ø) - an empty set. Its enumerated form (i.e. written out in full) is represented by a pair of curly brackets "{}".
A Universal set (ξ) contains all of the elements that are existent / possible in the specific example (e.g. people in the world).
Subsets are sets which only contain values which are present in the set which it is a subset of. For example, B=A (means all of B are in A). Also, it is worth noting that N is a subset of Z which is a subset of R.
A "Proper Subset" is where there are fewer elements in the subset than the set which it is a subset of. Most subsets are proper subsets.
Elements within sets can be ordered in any way and will still represent the same set (e.g. {a,b,c} = {b,a,c}). Also, repeated elements in sets are irrelevant and can be taken out therefore {a,b,b} = {a,b} for example. The cardinality of a set is the number of elements in the set and has the notation #(X) where X is the set.
Some set operations that can be carried out upon multiple sets are as follows:
- Union - A u B (or)
- Intersection - A n B (and)
- Complement - ¬B (not)
- Difference - A-B or A/B (in A but not in B)
Venn Diagams are a graphical representation of sets involving overlapping ovals containing the elements listed in the sets. Different parts of the diagram can be shaded to show which bits are represented by combinations of sets and operations.
Laws:
- Double complement (negative) Law - the double negation of a set gives the original set. e.g. ¬¬A = A
- Idempotent Law - a set associated with itself results in the same original set. e.g. A n A = A and A u A = A
- Commutative Law - sets can be swapped around freely in a simple association. e.g. A n B is the same as B n A
- Associative Law - with operators of same type being used, brackets can go anywhere. e.g. (A n B) n C is the same as A n (B n C)
- Distributive Law - a set that is related to sets within a pair of brackets can be directly related to each set within the brackets. e.g. A n (B u C) is the same as (A n B) u (A n C)
- Identity Law - where a set is associated with another set but the original set is still the result. e.g. A n ξ = A and A u Ø = A
- Annihilation Law - where a set is associated with another set and the original set disappears from the resulting set formula. e.g. A u ξ = ξ and A n Ø = Ø
- Inverse Law - a set associated with its inverse will result in either Ø or ξ. e.g. A n ¬A = Ø and A u ¬A = ξ
Search for "Set Representation" on:
Google |
Kelkoo |
Amazon |
eBay (UK) |
eBay (US)
Search for "Set Representation" on the rest of Computing Students: Set Representation
|
