Power set

The concept in issue here is power sets and universal sets and how these are best understood can be explained by real-life exercises. Imagine a small garden with three types of plants: It includes roses, tulips, and daisies that can be red, white, black, pink, and black. The power set incorporates everything that has been noted above, including the void, every plant as an individual power set member, combinations of two plants, and all three plants as the power set members. On the other hand, the universe 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 power sets and 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. Over the last ten years of the JEE Main exam (from 2013 to 2023), a total of three questions have been asked on this concept, including one in 2014, one in 2022, and one in 2023.

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 particularly useful 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…...

Define Power set

A power set is the set of all possible subsets of a given set, including the empty set and the set itself. The power set always contains 2ⁿ elements, where n is the number of elements in the original set.

The collection of all subsets of a set A is called the power set of A. It is denoted by P(A).

For Example if set A = {a, b, c}, then

P(A) = {φ, {a}, {b}, {c}, {a, b}, {b, c}, {c, a}, {a, b, c}}

• The power set of any set is non-empty.

• Each element of a Power set is a set.

• Number of elements in P(A) = Number of subsets of set A = 2^{Number of elements in set A}

Properties of Power Set

• It is much larger than the original set.
• The number of elements in the power set of A is 2ⁿ, where n is the number of elements in set A
• The power set of a countable finite set is countable.
• For a set of natural numbers, we can do one-to-one mapping of the resulting set, P(S), with the real numbers.
• P(S) of set S, if operated with the union of sets, the intersection of sets and complement of sets, denotes the example of Boolean Algebra.

Solved Examples Based On the Power Set and the Universal Set:

Example 1: Power set of {1,2,3} is

1)

2)

3)

4) None of the above

Solution:

As we learned

In this Question,

Option (3) shows all the subsets of {1,2,3}

Hence, the answer is the option 3.

Example 2: Find the number of elements in the power set of {1,2,3,4}.

1) 8

2) 16

3) 32

4) 64

Solution

As we learned

In this Question,

The power set of {1,2,3,4} will have elements.

Hence, the answer is the option 2.

Example 3: 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.

Example 4: Let Then the number of elements in the set is__________.

Solution:

Total subsets of

Subsets of such that

Subsets of

Required Subsets

Hence, the answer is (112).

Summary

Power sets and universal sets are some of the most important principles in the field of set theory and facilitate the determination of the links among various sets of elements. This means that the power set contains all the possible subsets of a particular set, depicting all the possibilities in the certain range of the set in question. On the other hand, the universal set contains all the elements that exist in a given context, hence, it provides a broad view that encompasses all elements within a given context.