Venn Diagrams and Union of Sets: Class 11 Maths Explained Step by Step

Venn Diagrams and Union of Sets: Class 11 Maths Explained Step by Step

Once you can write sets and compare them as subsets, Class 11 Maths introduces a visual tool — the Venn diagram — and the first proper operation performed on two sets: the union. Together, these give you both a picture and a formula for combining sets, which you will use repeatedly in probability and relations later in the syllabus.

What Is a Venn Diagram?

A Venn diagram is a way of representing sets and the relationships between them using rectangles and closed curves, usually circles. Venn diagrams are named after the English logician John Venn (1834–1883).

The convention is simple:

• The universal set U is represented by a rectangle.
• Individual sets (subsets of U) are represented by circles inside the rectangle.
• The elements of each set are written as points inside their respective circle.

For example, if U = {1, 2, 3, …, 10} is the universal set and A = {2, 4, 6, 8, 10}, then A is drawn as a circle inside the rectangle, with the numbers 2, 4, 6, 8, 10 marked inside it and the remaining numbers (1, 3, 5, 7, 9) marked outside the circle but still inside the rectangle.

If a set B is a subset of A (say B = {4, 6}), the circle for B is drawn entirely inside the circle for A — visually showing that B ⊂ A.

Venn diagrams become especially useful once you start performing operations on sets, because they let you see the result of an operation instead of just calculating it.

The Union of Two Sets

The union of two sets A and B is the set containing all elements that are in A, or in B, or in both. It is written A ∪ B and read as "A union B."

Formally: A ∪ B = {x : x ∈ A or x ∈ B}

The key rule to remember: any element that is common to both sets is written only once in the union, never twice.

Worked Example

Let A = {2, 4, 6, 8} and B = {6, 8, 10, 12}.

A ∪ B = {2, 4, 6, 8, 10, 12}

Notice that 6 and 8 appear in both A and B, but they are listed only once in the final union.

Union with a Subset

If B is a subset of A, then A ∪ B = A. This makes sense intuitively: since every element of B is already inside A, combining them adds nothing new.

For example, if A = {a, e, i, o, u} and B = {a, i, u}, then A ∪ B = {a, e, i, o, u} = A, because B ⊂ A.

Real-World Style Example

Suppose X = {Ravi, Meera, Karan} is the set of students on the school hockey team, and Y = {Meera, Sameer, Arjun} is the set of students on the football team. Then:

X ∪ Y = {Ravi, Meera, Karan, Sameer, Arjun}

This represents every student who plays hockey, football, or both — a very natural, real-world reading of "union."

Representing Union with a Venn Diagram

In a Venn diagram, A ∪ B is shown by shading both circles completely — every region that belongs to A, to B, or to their overlap. This is the visual equivalent of "everything in A or B."

Properties of Union

The union operation follows several algebraic laws that are useful for simplifying expressions and proving results:

1. Commutative law: A ∪ B = B ∪ A — order does not matter.
2. Associative law: (A ∪ B) ∪ C = A ∪ (B ∪ C) — grouping does not matter.
3. Identity law: A ∪ φ = A — combining with the empty set changes nothing.
4. Idempotent law: A ∪ A = A — combining a set with itself gives the same set back.
5. Law of U: U ∪ A = U — combining any subset with the universal set gives the universal set.

These properties often appear as short "prove that" questions in exams, so it's worth memorising each one alongside a quick example.

Common Mistakes to Avoid

• Listing common elements twice in the union — remember, each element appears only once, no matter how many original sets it belonged to.
• Confusing union with intersection — union collects everything from both sets, while intersection (covered separately) keeps only what's common. Mixing the two symbols, ∪ and ∩, is one of the most frequent errors in this chapter.
• Forgetting that A ∪ φ = A, not φ — the empty set adds nothing but also removes nothing.

Solved Examples

Example 1: If A = {1, 3, 5} and B = {1, 2, 3}, find A ∪ B.
Solution: A ∪ B = {1, 2, 3, 5}.

Example 2: If A and B are two sets such that A ⊂ B, what is A ∪ B?
Solution: Since every element of A is already in B, A ∪ B = B.

Example 3: If A = {1, 2, 3}, B = φ, find A ∪ B.
Solution: A ∪ B = {1, 2, 3} = A, illustrating the identity law.

Keep Learning

• Equal Sets and Subsets in Class 11 Maths: Definitions, Rules & Examples
• Intersection and Difference of Sets: Class 11 Maths Concepts with Examples
• Complement of a Set and De Morgan's Laws: Class 11 Maths Guide

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