Universal set

Universal Sets is an important concept in mathematics, central to various fields such as statistics, geometry, and algebra. Imagine a small garden with three types of plants: Roses, tulips, and daisies that can be red, white, black, pink, and black. This universal set refers to the totality of the plant species in the garden. Such plants include roses, tulips, daisies, and any other plant.

In this article, we will cover the concept of universal sets. This concept falls under the broader category of sets relation and function, a crucial Chapter in class 11 Mathematics. It is not only essential for board exams but also for competitive exams like the Joint Entrance Examination (JEE Main), and other entrance exams such as SRMJEE, BITSAT, WBJEE, BCECE, and more.

Define Set

Sets are a foundational concept in mathematics, central to various fields such as statistics, geometry, and algebra. A set is simply a collection of distinct objects, considered as a whole. These objects, called elements or members of the set, can be anything: numbers, people, letters, etc. Sets are instrumental in defining and working with groups of objects that share common properties.

It is a well-defined collection of distinct objects and it is usually denoted by capital letters A, B, C, S, U, V…..

Universal set

A set that contains all sets in a given context is called the “Universal Set”. The universal set is usually denoted by U, and all its subsets are denoted by the letters A, B, C, etc.

For example, for the set of all integers, the universal set can be the set of rational numbers or, for that matter, the set R of real numbers.

If A is a set of all tigers in a jungle, and B is a set of all deers in the jungle, then the universal set can be all the animals of that jungle, as all tigers and all deers are subsets of this set.

This basic set is called the “Universal Set”.

All the sets under consideration are likely to be subsets of a set called the universal set which is denoted by

Ex- The set of all letters in the alphabet of the English language U = {a,b,c,.......,x,y,z} is the universal set of vowels in the alphabet of the English language. i.e. A={a,e,i,o,u}

Properties of Universal Set:

1. When we take union from a universal set then a universal set will come.U A = U

Note: If A is a subset of B, then A ∪ B = B

2. The universal set contains all elements under consideration

Complement of a set

Let U be the universal set and A is a subset of U. Then the complement of A is the set of all elements of U which are not the elements of A.

Symbolically, we use A' or A^c to denote the complement of A with respect to U.

A' = {x∶ x ∈ U and x ∉ A }. Obviously, A' = U – A

The complement of the universal set is the empty set.

Summary: Universal sets play a crucial role in both computational and theoretical contexts due to their fundamental nature in mathematics, which makes defining collections of objects, performing operations like unions and intersections, and establishing relationships more efficient.

Solved Examples Based On the Universal Set:

Example 1: If A is any set, then which of the following is not a property of a Universal Set?

1)

2)

3)

4)

Solution

As we learned

There is no common portion between A and A', hence A A' = . So (2) is wrong.

Hence, the answer is the option 2.

Example 2: If G={-9,-8,-7,-6} and {8, 2, 7, 4}, then which of the following MAY BE a universal set?

1) Set of all whole numbers

2) Set of all irrational numbers

3) Set of all integers

4) All of the above

Solution

We know the elements of both sets G and H are there in the set of all integers, hence option (3) can be a universal set.

Hence, the answer is the option 3.