Universal set

Consider the set of all natural numbers. The possible universal set for the set of all natural numbers can be integers or real numbers. An Universal Set is set that includes all the possible elements.

Universal sets play a crucial role in defining collections of objects, performing operations like unions and intersections, and establishing relationships, making them more efficient.

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.

This Story also Contains

1. Universal set Definition
2. Properties of Universal Set
3. Solved Examples Based On the Universal Set

Universal set Definition

A set that contains all the elements or objects of other related sets including its own elements is called the Universal Set. The symbol of universal set is usually denoted by $U$, and all its subsets are denoted by the letters $\mathrm{A}, \mathrm{B}, \mathrm{C}$, etc.

Universal set 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.

Let's look into some universal set example in real life.

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.

Similarly, if $A$ is a set of all roses and $B$ is a set of all tulips, then the universal set could be the set of all flowers in the world where the sets $A$ and $B$ are the subsets of the universal 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 $\Omega$ or $S$ or $U$.

Example: 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. $A=\{a,e,i,o,u\}$

Venn Diagram of Universal Set

Let $U = \{1,2,3,4,5,6,7,8,9,10,11,12,13,14,15\}$ and $A = \{10,4,2,6,1,9,3\}$, $B=\{4,13,1,9,3,15,14\}$ and $C =\{2,6,1,9,3,14,15,11,12\}$.

The venn diagram to represent the universal set and its subsets of the above example is

Properties of Universal Set

1. When we take union from a universal set then the result will be the universal set. $U \cup A=U$

Note: If $A$ is a subset of $B$, then $A \cup B=B$
2. The universal set contains all elements under consideration

Complement of Universal 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 $A$.

Complement of set $A$ is denoted by $A^\prime$ or $A^C$

$
A^{\prime}=\{x: x \in U \text { and } x \notin A\} \text {. Obviously, } A^{\prime}=U-A
$
The complement of the universal set is the empty set (i.e) $U^{\prime}=\phi$

Universal Set and Union of Set

The universal set is the union of all its subsets together with some more elements in some cases. Now, let us look into an example. Consider three sets with elements set $A = \{a, b, c\}$ and set $B = \{e, f, g\}$. Let's find all the possible universal set for the sets $A$ and $B$.

1. The universal set of the above two sets can be the set of all alphabets $\{a, b, c, d, …, z\}$
2. The union between $A$ and $B$ is given as: $A ∪ B = \{a, b, c, e, f, g\}$

Thus, we can see that the universal set contains the elements from $A, B,$ and $U$ itself, whereas the union of $A$ and $B$ contains elements from only $A$ and $B$.

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) $A \cup A^c=U$
2) $A \cap A^c=U$
3) $U-A=A^c$
4) $U-A^c=A$

Solution

There is no common portion between $A$ and $A^{\prime}$,

$A\cap A^{c}=\phi$. So $(2)$ is wrong.

Hence, the answer is the option 2.

Example 2: If $\mathrm{G}=\{-9,-8,-7,-6\}$ 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 3: How many elements are in the complement of $A$, designated as $\mathrm{A}^{\prime}$, if set $A$ contains $13$ elements and the universal set $U$ has $20$ ?

1) $12$
2) $7$
3) $20$
4) $0$

Solution

All the elements in the universal set $U$ that aren't in set A are found in the complement of set $A$.

$
\therefore n\left(A^{\prime}\right)=n(U)-n(A)=20-13=7
$
Hence, the correct answer is $7$.

Example 4: Which of the following can be the universal set if two of the sets involved are
$A$ = {Set of Indians born after 1999}
$B$ = {Set of Indian born after 2000}

1) {Set of Indians born after 2002}
2) {Set of Indian born before 2002}
3) {Set of Indian born before 2005}
4) {Set of Indian born after 1998}

Solution

As the set of people born after $1998$ will also contain people born after $1999$ and $2002$. So, it is the universal set.

Hence, the answer is the option (4).

Example 5: The set which contains those elements of Universal set, which are not contained in set $A$, is called complement of a set $A$. (True/False)

Solution

Complement of a Set:

If $U$ is a universal set and $A$ is a subset of $U$, then the complement of $A$ is the set which contains those elements of $U$, which are not contained in $A$. Symbolically, we write $A′$ to denote the complement of $A$ with respect to $U$.

The statement is true.

List of Topics Related to Universal Set