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:
- Roster Form: Writing all the elements explicitly. Example: U={1,2,3,4,5}
- Set-builder Form: Describing the property of elements. Example: U={x:x∈N,x<6}
- 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
- Every set is a subset of the universal set: A⊆U
- Complement of a set is always defined with respect to the universal set: A′=U−A
- The universal set depends on context. It changes according to the situation.
- 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=N ) and ( B = {x:x is a multiple of 3, x≤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≤15 ))
Question 3 : If ( U=R ), ( A = {x∈R:x2<1} ), find ( A' ).
- ( A = (-1, 1) )
- ( A' = U - A = (−∞,−1]∪[1,∞))
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∪B∣=∣A∣+∣B∣−∣A∩B∣=40 )
- (25+18−∣A∩B∣=40⟹∣A∩B∣=3)
Question 5 : If the universal set is ( U = {a, b, c, d} ), how many subsets does ( U ) have?
- Number of subsets = ( 24=16 )
8.0Practice Questions on Universal Sets
- If U={1,2,3,4,5,6,7,8},A={2,4,6,8}, find A'
- Let U={x:x≤10,x∈N},A={2,4,6,8,10}. Find n(A)+n(A′)
- If U={1,2,3,4,5,6},A={1,2,3},B={3,4,5}. Find (A∪B)'
- 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=R, and A={x∈R:x2<4} find A'