Intersection and Difference of Sets: Class 11 Maths Concepts with Examples

Intersection and Difference of Sets: Class 11 Maths Concepts with Examples

After union, Class 11 Maths introduces two more set operations that are just as important: intersection, which finds what two sets share, and difference, which finds what belongs to one set but not the other. Both are tested heavily through Venn diagram questions and multi-set word problems, so getting comfortable with their exact definitions matters.

What Is the Intersection of Two Sets?

The intersection of two sets A and B is the set of all elements that are common to both A and B. It is written A ∩ B, read as "A intersection B."

Formally: A ∩ B = {x : x ∈ A and x ∈ B}

The key word here is "and" — an element must belong to A and B simultaneously to be included in the intersection. Compare this with union, where the key word is "or."

Worked Example

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

A ∩ B = {6, 8}

Only 6 and 8 appear in both sets, so only they are included.

Intersection with a Subset

If B ⊂ A, then A ∩ B = B. This is because every element of B is already inside A, so the "common" elements are exactly the elements of B itself.

For example, if A = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10} and B = {2, 3, 5, 7}, then A ∩ B = {2, 3, 5, 7} = B, confirming B ⊂ A.

Disjoint Sets

If two sets A and B have no elements in common, so that A ∩ B = φ, they are called disjoint sets.

For example, A = {2, 4, 6, 8} and B = {1, 3, 5, 7} are disjoint sets, since no number appears in both lists. In a Venn diagram, disjoint sets are drawn as two separate, non-overlapping circles.

Representing Intersection with a Venn Diagram

In a Venn diagram, A ∩ B is shown by shading only the overlapping region between the two circles — the part that belongs to both A and B simultaneously.

Properties of Intersection

Intersection follows several algebraic laws, similar in style to the laws for union:

1. Commutative law: A ∩ B = B ∩ A
2. Associative law: (A ∩ B) ∩ C = A ∩ (B ∩ C)
3. Law of φ and U: φ ∩ A = φ, and U ∩ A = A
4. Idempotent law: A ∩ A = A
5. Distributive law: A ∩ (B ∪ C) = (A ∩ B) ∪ (A ∩ C) — intersection distributes over union, much like multiplication distributes over addition in arithmetic.

The distributive law is a frequent "prove that" question, and it is easiest to verify using a Venn diagram, shading each side of the equation separately to confirm they match.

What Is the Difference of Two Sets?

The difference of sets A and B, in that specific order, is the set of elements that belong to A but not to B. It is written A − B, read as "A minus B."

Formally: A − B = {x : x ∈ A and x ∉ B}

Important: A − B is generally not equal to B − A. Order matters here, unlike in union and intersection.

Worked Example

Let A = {1, 2, 3, 4, 5, 6} and B = {2, 4, 6, 8}.

• A − B = {1, 3, 5} — elements in A that are not in B.
• B − A = {8} — the only element in B that is not in A.

Clearly A − B ≠ B − A, confirming that difference is not commutative.

Representing Difference with a Venn Diagram

A − B is shown by shading the part of circle A that does not overlap with circle B. This visually confirms why A − B and B − A give different results — you are shading a different "leftover" region each time.

A Useful Fact

The three sets A − B, A ∩ B, and B − A are always mutually disjoint — meaning no two of them ever overlap. Together, they neatly partition the union A ∪ B into three separate, non-overlapping pieces.

Solved Examples

Example 1: Let V = {a, e, i, o, u} and B = {a, i, k, u}. Find V − B and B − V.
Solution: V − B = {e, o} (elements in V not in B). B − V = {k} (the element in B not in V).

Example 2: If A = {3, 5, 7, 9, 11} and B = {7, 9, 11, 13}, find A ∩ B.
Solution: A ∩ B = {7, 9, 11}.

Example 3: Are {1, 2, 3, 4} and {x : x is a natural number, 4 ≤ x ≤ 6} disjoint?
Solution: The second set is {4, 5, 6}. Since 4 is common to both sets, they are not disjoint.

Common Mistakes to Avoid

• Reversing the order in difference: A − B and B − A are different sets — always match the order to the question.
• Confusing φ with disjoint: disjoint sets have an intersection equal to φ, but neither set itself needs to be empty.
• Forgetting that intersection needs "and," not "or" — mixing this up with union is one of the most common errors in this chapter.

Keep Learning

• Venn Diagrams and Union of Sets: Class 11 Maths Explained Step by Step
• Complement of a Set and De Morgan's Laws: Class 11 Maths Guide
• Equal Sets and Subsets in Class 11 Maths: Definitions, Rules & Examples

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