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**Set Theory Concepts**

∙ Introduction

∙ Definitions

∙ Universe & Subsets

∙ More on subsets

∙ Operations on sets

∙ Properties of operations

∙ Set graphs & Line graphs

∙ Introduction

∙ Definitions

∙ Universe & Subsets

∙ More on subsets

∙ Operations on sets

∙ Properties of operations

∙ Set graphs & Line graphs

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.

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**.

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?

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.*

Since,

every circle in C is a circle in C!

By definition, C ⊂ C

Therefore,

**(1) every set is a subset of itself.**

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.

Thus,

**(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.