To define a universal set (usually denoted as U), the field of mathematics portrays a variety of applications to understand its usage. It is a set consisting of all the elements present in other given sets. A group of elements or objects are known as a Set. A universal set can be represented in the form of a Venn diagram. 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.
Various types of sets are formed based on the universal set elements and the criteria that satisfy its 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,
S = {x I x ∈ N and 7 < x < 8} = ∅
Definite elements present in a set is known as a finite set. For example,
S = {x I 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 I 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.
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.
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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.
A ⊄ B indicates that the elements of A are not there in B.
The universal set includes all elements, but the union of 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}
A subset is defined as a set of all elements present in one 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.
Q1. What is a Superset and How is it Related to a Universal Set?
When we consider Set A is a subset of B, then B is known as a superset of A. This indicates the set B has all the elements specifically found in set A and nowhere else. Therefore, sets A and B are equal. When they are unequal, then Set A would be a proper set of B. This phenomenon is termed an inclusion.
For example, if natural numbers are also integers. Let’s consider sets N and Z that represent all the natural numbers and integers respectively, then we can denote as, N ⊂ Z
Where, N is a proper subset of Z, and therefore, Z is called the superset of N.
Q2. What are the Applications of a Universal Set?
An organized collection of elements, objects, or data with certain criteria followed is termed as a set. Usually, sets are represented in the roster or set builder form. The objects present within these sets are known as elements and they can form subsets from the original data. Classifying objects is a fundamental activity each of us carries out in our daily lives.
Organizing kitchen cutlery, crockery, and utensils can be known as forming sets. Buying certain supplies or grocery items can also be known as categorizing based on the item bought. Music playlists can be enclosed with a variety of songs separated based on genre, artist, or mood. Rules followed by many communities and institutions are particularly adapted to their lifestyle.
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