Elementary Concepts of Set Theory

Idea of Universe and Subsets

English Letters Almost all the sets considered are members of a larger group


For instance,


{a, b, c, d, e} is in a larger group of (English) letters.



The term to identify the larger group is universal set or universe

It means,


all the things now under consideration

The counterpart to the idea of universal set is the idea of subset.



abc subset Within every sets considered, smaller sets can be found.


For instance,


{a, b, c} is the smaller set in the larger {a, b, c, d, e}


Mathematically, {a, b, c} ⊂ {a, b, c, d, e}


A subset is defined as,


a set in which every member is also a member of the original set.



set of circles If the set of circles, C is a subset taken from the universal set (U) composed of triangles and circles.


How many sets can be considered subsets of C?


subset of circles An obvious subset is the set of concentric circles {2, 4, 7}. Similarly, set of first three circles {2, 3, 4} and set of non-concentric circles {3, 6, 9} are also subset of C.


Starting with a set with only one element and then considering every combination, the number of subsets of C is obviously a larger number, but finite.


There are two special subsets.



subset of itself Since,

every circle in C is a circle in C!


By definition, C ⊂ C



(1) every set is a subset of itself.




Empty set Let us consider from set C,

set of purple circles.


Since the set of purple circles is considered under the context of set C,

the set of purple circles is a subset of C.


However, there are no purple circles in C. That is,


set of purple circles (in C) is an empty set (denoted ϕ) also called a null set.



(2) an empty set is a subset. Therefore, ϕ ⊂ C.




Similarly, set of squares, set of spheres, set of polygons, set of apples etc. are all empty set, i.e., = ϕ.


Being empty set, they are all therefore subset of set of circles, C.