Sets:
pure set: a set containing scalars only
empty set: pretty vacant
higher-order set: a set containing set(s)
mixed set:a set known to contain at least one scalar AND at least one set
universal set: set containing everything, the universe
Types:
Ordered pair or pair
Powerset
Cartesian product or cross product
Partitioning
Bell Numbers
wrt cardinality:
0: Empty set
1: Singleton set or unit set
2: Unordered pair
Finite set
Infinite set
Universal set
wrt relations:
Equal sets
Equivalent sets
Overlapping sets
Disjoint sets
wrt set operations:
Union
Intersection
Difference
Relative complement
Properties:
Commutative
Associative
Distributive
Idempotency
Identity
Transitive
Involution
De Morgan's Law
Closure
Types of Sets
Sets can be classified into many types, including: finite, infinite, universal, singleton, empty set.
Proper Subset A Set X is a proper subset of set Y (Written as X⊂Y) if every element of X is an element of set Y and |X|<|Y|.
X={1,2,3,4,5,6} and Y={1,2}. Here set Y⊂X since all elements in Y are contained in X too and X has at least one element is more than set Y.
Universal Set It is a collection of all elements in a particular context or application. All the sets in that context or application are essentially subsets of this universal set. Universal sets are represented as U.
Example − We may define U as the set of all animals on earth. In this case, set of all mammals is a subset of U, set of all fishes is a subset of U, set of all insects is a subset of U, and so on.
Singleton Set or Unit Set Singleton set or unit set contains only one element. A singleton set is denoted by {s}. Example − S={x|x∈N, 7<x<9} = {8}
Equal Set If two sets contain the same elements they are said to be equal. Example − If A={1,2,6} and B={6,1,2}, they are equal as every element of set A is an element of set B and every element of set B is an element of set A.
Equivalent Set If the cardinalities of two sets are same, they are called equivalent sets. Example − If A={1,2,6} and B={16,17,22}, they are equivalent as cardinality of A is equal to the cardinality of B. i.e. |A|=|B|=3
Overlapping Set Two sets that have at least one common element are called overlapping sets. In case of overlapping sets − n(A∪B)=n(A)+n(B)−n(A∩B) n(A∪B)=n(A−B)+n(B−A)+n(A∩B) n(A)=n(A−B)+n(A∩B) n(B)=n(B−A)+n(A∩B) Example − Let, A={1,2,6} and B={6,12,42}. There is a common element ‘6’, hence these sets are overlapping sets.
Disjoint Set Two sets A and B are called disjoint sets if they do not have even one element in common. Therefore, disjoint sets have the following properties − n(A∩B)=∅ n(A∪B)=n(A)+n(B) Example − Let, A={1,2,6} and B={7,9,14}, there is not a single common element, hence these sets are overlapping sets.
Power set of a set S is the set of all subsets of S including the empty set. The cardinality of a power set of a set S of cardinality n is 2^n
Set Operations
Union
Intersection
Difference
Complement
Cartesian (cross) product
Partitioning
Partition of a set, say S, is a collection of n disjoint subsets, say P1,P2,…Pn that satisfies the following three conditions −
Pi does not contain the empty set.
[Pi≠{∅} for all 0<i≤n] The union of the subsets must equal the entire original set.
[P1∪P2∪⋯∪Pn=S] The intersection of any two distinct sets is empty.
[Pa∩Pb={∅}, for a≠b where n≥a,b≥0] Example
Let S={a,b,c,d,e,f,g,h} One probable partitioning is {a},{b,c,d},{e,f,g,h} Another probable partitioning is {a,b},{c,d},{e,f,g,h}
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