[Maths Class Notes] on Universal Set Pdf for Exam

A set is actually any given set of objects or elements. A universal set is a collection of elements of all related sets and subsets including itself. It is a set consisting of all the elements present in other given sets.It is represented by the symbol U usually and sometimes by E also. E.g. All the planets in the solar system are a good example of the Universal set. The universal set can be finite or infinite.

 

The universal set can be diagrammatically represented by a Venn diagram. In a Venn diagram, the Universal set is represented by a rectangle while its subsets by circles. Universal sets help to define the bulk of data or a set of elements of commonality to make it easier to segregate and arrive at a suitable conclusion. There is a difference that exists between the Universal set and Union set. The difference is mainly because the Union set contains all the elements of the related subsets but not any other elements than that while the Universal set includes all elements of the subsets and the elements of itself also. 

Types of Sets

Various types of sets are formed based on the universal set elements and the criteria that satisfy their place in a new set. Some of them are as follows:

It is denoted by ∅. The elements in an empty set are finite, the empty set is finite. The direction of the empty set or null set is zero. For example, 

X = {x : x is a prime number and 14

Definite elements present in a set is known as a finite set. For example, 

S = {x : x ∈ N and 70 > x > 50}

An infinite group of elements present in a set is known as an infinite set. For example,  S = {x : x ∈ N and  x > 10}

If the directions of two sets are the same, they are known as equivalent sets. For example, If A = {1, 2, 6} and B = {16, 17, 22}, they are equivalent to the cardinality of A is equal to the cardinality of B. i.e. ।A। = ।B। = 3

Set X is a subset of set Y (Written as X ⊆ Y) if every element of X is an element of set Y.

 Universal Set Notation

The universal set has no standard notation provided, but it can be denoted by the symbol’s ‘U’, ‘V’ or ‘ξ’

The set notation is usually indicated by using curly brackets, {} and each element in the set is separated by commas like {4, 7, 9}, where 4, 7, and 9 are the elements of sets. A Venn diagram of a universal set is represented in the form of two circles enclosed in a box known to be the universal set.

Define Universal Set With An Example

When considering two sets, A = {1,2,3} and B = {1,a,b,c}, the universal set is composed of the following elements, U = {1,2,3,a,b,c}.

When A = {1, 2, 3}, B = {{1, 2, 3}, 4, 5} and C = {{1, 2, 3}, 4, 5, 6, 7} are given, then B ⊂ C indicates all the elements of set B are also the elements of C.

 

Universal Sets and Union of a Set

The universal set includes all elements, but the union of the two sets states that A and B have combined their elements. The union set operand is indicated as ‘∪’.

For instance, 

Set A = {a,b,c} and set B = {c, d, e} and U = {1, 2}. 

Therefore, the universal set for set A, B, and U itself will be;

U = {a,b,c,d,e,1,2}

The union of set A and B is indicated as,

A U B = {a,b,c,d,e}

Universal Sets and Subsets

A subset is defined as a set of all elements present in one that is also present in another set. For instance, taking two sets X and Y, the elements of set X are also present in set Y. Hence, Set X is a subset of Y.

The definition of a subset can be represented as,

a ∊ A and a ∊ B, then A⊂B (where ‘⊂’ means ‘subset of’). 

The opposite of this case is also true,

When, A ⊂ B and a ∊ A, then a ∊ B.

When A is not a subset of B, it is represented by A ⊄ B. 

So, A ⊂ B, not all elements of B will be in set A. 

However, if A ⊂ B and B ⊂ A, implies that A = B.

This is represented by:

A ⊂ B and B ⊂ A ⇔ A = B

Therefore, ⇔ represents if and only if.

This is all about the universal set and its different types with proper explanation. Understand the difference between these types to grab hold of the concept and to solve questions on sets easily.