Set Theory Part (1)

A sets is collection of well-defined entities, objects or elements and also describing by listing elements, separated by commons within brackets.

TYPES OF SETS.

Universal sets are 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.

Null/Empty sets is an empty set contains no elements. It is denoted by ∅. As the number of elements in an empty set is finite, empty set is a finite set. The cardinality of empty set or null set is zero.

Example:

{ } = ∅

A= {1,3,5,7}

B= {2,4,6,8}

A⋂ B= { } = ∅

Equal sets are 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 sets are If the cardinalities of two sets are same.

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.

|A| = |B| = 3

Subset sets is a set Z is a subset of set X (Written as X ⊆ Z) if every element of X is an element of set Z.

Example:

Let, X = {1, 2, 3, 4, 5, 6} and Z = {1, 2}. Here set X is a subset of set Z as all the elements of set X is in set Z. Hence, we can write X ⊆ Z.

Finite sets are if its element can be counted and the process terminates at a certain natural number.

Example:

 {1,2,3,4,5}          OR       {1,2,3,4,5………….100}

Infinite sets are a set in which the process of counting does not terminate.

Example:

Set of natural  OR

{1,3,5,7,9…….} set of odd number OR

{2,4,6,8,10……} set of even number.

NOTATION SETS

{ }	Empty/Null sets
∪	Universe sets
A ═ B	Set of equivalence
A ⊆ B	Subset
A ⊂ B	Proper set
A ∪ B	Union
A ⋂ B	Intersection

Example 1:

X = {a,e,i,o,u} Y= {o,h,f,e}

A ∪ B = {a,e,i,o,u,h,f}

A ⋂ B = {o,e}

Example 2:

Z = {2,4,6,8} W = {1,3,5,7}

A ∪ B = {1,2,3,4,5,6,7,8}

A ⋂ B = { } = ∅