Universal Set

1.0What is a Universal Set?

A Universal Set is the set that contains all elements under consideration for a particular discussion or problem.

It is denoted by : U

Mathematically,

U={all possible elements relevant to the context}

For example, if the discussion is about natural numbers less than 10:

U={1,2,3,4,5,6,7,8,9}

2.0Symbol of Universal Set

• Universal sets are generally denoted by U.
• In Venn diagrams, it is represented by a rectangle that encloses all sets inside it.

3.0Representation of Universal Set

A universal set can be represented in three ways:

1. Roster Form: Writing all the elements explicitly. Example: U={1,2,3,4,5}
2. Set-builder Form: Describing the property of elements. Example: U={x:x∈N,x<6}
3. Venn Diagram Representation

• Universal set is drawn as a rectangle.
• Other sets are drawn as circles inside the rectangle.

Example: If U={1,2,3,4,5}, and A={2,4}, then A is shown as a circle inside rectangle U.

4.0Examples of Universal Set

Example 1: Numbers

If we talk about even numbers less than 20, the universal set can be all natural numbers less than 20:

U={1,2,3,…,19}

And A={2,4,6,…,18}.

Example 2: Geometry

In the context of triangles in geometry,

U={all types of triangles: equilateral, isosceles, scalene, right-angled}

Example 3: JEE-Oriented Example

If the discussion is about solutions of quadratic equations, the universal set can be the set of complex numbers (C).

5.0Properties of Universal Set

1. Every set is a subset of the universal set: A⊆U
2. Complement of a set is always defined with respect to the universal set: A′=U−A
3. The universal set depends on context. It changes according to the situation.
4. If U is finite with n(U) elements, then: n(A)+n(A′)=n(U)

6.0Universal Set in Venn Diagrams

• The universal set U is represented by a rectangle.
• Subsets are represented by circles inside the rectangle.
• Operations like union (∪), intersection (∩), and complement (A′) are all defined with respect to U.

Example: If U={1,2,3,4,5}, and A={1,2},B={3,4}, then A∪B={1,2,3,4}, and A′={3,4,5}.

7.0Solved Examples on Universal Sets

Question 1: If ( U = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10} ) and ( A = {2, 4, 6, 8, 10} ), find ( A' ).

• ( A' = U - A = {1, 3, 5, 7, 9} )

Question 2 : Let ( $U=\mathbb{N}$ ) and ( B = {$x:x\text{ is a multiple of 3, }x\le 15$} ). List elements of ( B ) and ( B' ) (up to 15).

• ( B = {3, 6, 9, 12, 15} )
• ( B' = U - B = {1, 2, 4, 5, 7, 8, 10, 11, 13, 14} ) (for ( $x\le 15$ ))

Question 3 : If ( $U=\mathbb{R}$ ), ( A = {$x\in \mathbb{R}:x^2<1$} ), find ( A' ).

• ( A = (-1, 1) )
• ( A' = U - A = $(-\infty ,-1]\cup [1,\infty )$)

Question 4 : In a class of 40 students (( U )), 25 like mathematics (( A )), and 18 like physics (( B )). If each student likes at least one subject, how many like both?

By inclusion-exclusion:

• ( $|A\cup B|=|A|+|B|-|A\cap B|=40$ )
• ($25+18-|A\cap B|=40⟹|A\cap B|=3$)

Question 5 : If the universal set is ( U = {a, b, c, d} ), how many subsets does ( U ) have?

• Number of subsets = ( $2^4=16$ )

8.0Practice Questions on Universal Sets

1. If $U=\{1,2,3,4,5,6,7,8\},A=\{2,4,6,8\}$, find A'
2. Let $U=\{x:x\le 10,x\in \mathbb{N}\},A=\{2,4,6,8,10\}$. Find $n(A)+n(A^{\prime })$
3. If $U=\{1,2,3,4,5,6\},A=\{1,2,3\},B=\{3,4,5\}$. Find $(A\cup B)$'
4. In a survey of 100 students, 60 like maths, 45 like physics, and 25 like both. Represent using a Venn diagram with universal set U. 

5. If $U=\mathbb{R}$,  and $A=\{x\in \mathbb{R}:x^2<4\}$ find A'