Universal set

This is a basic set; in a particular context whose elements and subsets are relevant to that particular context. For example, for the set of vowels in English alphabet, the universal set can be the set of all alphabets in English. Universal set is denoted by 𝕌.

Example 1.
a) If we were discussing searching for books, the universal set might be all the books in
the library.
b) If we were grouping your Facebook friends, the universal set would be all your
Facebook friends.
c) If you were working with sets of numbers, the universal set might be all whole
numbers, all integers, or all real numbers

[Definition] The universal set, at least for a given collection of set theoretic computations, is the set of all possible objects.

If we declare our universal set to be the integers then {½,⅔}, is not a Well defined set because the objects used to define it are not members of the universal set. The symbols {½,⅔} do define a set if a universal set that includes ½ and ⅔ is chosen. The problem arises from the fact that neither of these numbers are integers. The universal set is commonly written 𝕌. Now that we have the idea of declaring a universal set we can define another operation on sets.

Universal Set

Usually, in a particular context, we have to deal with the elements and subsets of a basic set which is relevant to that particular context. For example, while studying the system of numbers, we are interested in the set of natural numbers and its subsets such as the set of all prime numbers, the set of all even numbers, and so forth. This basic set is called the “Universal Set”. The universal set is usually denoted by 𝕌, and all its subsets 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. For another example, in human population studies, the universal set consists of all the people in the world.

Universe?

In order to work with sets we need to define a Universal Set, 𝕌, which contains all possible elements of any set we wish to consider. The Universal Set is often obvious from context but on occasion needs to be explicitly stated.

For example, if we are counting objects, the Universal Set would be whole numbers. If we are spelling words, the Universal Set would be letters of the alphabet. If we are considering students enrolled in ASU math classes this semester, the Universal Set could be all ASU students enrolled this semester or it could be all ASU students enrolled from 2000 to 2018. In this last case, the Universal Set is not so obvious and should be clearly stated.

⛲ Question 1. What universal set(s) would you propose for each of the following?
(i) The set of right triangles
(ii) The set of isosceles triangles
✍ Solution:
(i) For the set of right triangles, the universal set can be the set of triangles or the set of polygons.
(ii) For the set of isosceles triangles, the universal set can be the set of triangles or the set of polygons or the set of two-dimensional figures.

Universal Set

A universal set is a set that contains all the elements we are interested in. This would
have to be defined by the context.
A complement is relative to the universal set, so A_∁ contains all the elements in the universal set that are not in A.
Example 2.
Suppose the universal set is 𝕌=all whole numbers from 1 to 9. If A={1,2,4}, then
A_∁={3,5,6,7,8,9}.
🌈 Which Sets below may be considered as Universal Set(s) for the previous Sets?