Operations on Sets – Union, Intersection, Commutative, Associative & Distributive Laws Explained | Math With Raabi

 

📘 Operations on Sets — Union, Intersection, Distributive & Associative Laws

📚 In set theory, the main operations are union, intersection and complement. These operations help us combine and compare sets. Knowing the formal laws — commutative, associative and distributive — makes it easier to simplify set expressions and verify equalities.

🧭 Before studying operations, make sure you have read the previous topic: Understanding Sets in Mathematics . This provides a clear introduction to sets and notation. 🔗

Union, Intersection and Complement — Quick Reminder

Intersection (A ∩ B): The set of elements common to both A and B.
Union (A ∪ B): The set of elements that are in A or in B (or in both).
Complement (A′) (with respect to universal set U): All elements of U that are not in A.

Notation reminder: Use ∪ for union and ∩ for intersection.

Remember (Intersection Tip):
When finding A ∩ B, it’s easier to write the first set (A) first and check which of its elements are also in B. This reduces unnecessary comparison and helps avoid mistakes. ✅

Commutative Laws

The commutative laws say that changing the order of sets does not change the result:

• A ∪ B = B ∪ A
• A ∩ B = B ∩ A

Example (Commutative laws)
Let A = {a, b, c, d} and B = {b, d, f}.

Union:
A ∪ B = {a, b, c, d, f}   and   B ∪ A = {a, b, c, d, f}.
Intersection:
A ∩ B = {b, d}   and   B ∩ A = {b, d}.
Both results match, so commutative laws are verified.

Associative Laws

Associative laws allow grouping of sets without changing the result when three sets are involved:

• A ∪ (B ∪ C) = (A ∪ B) ∪ C
• A ∩ (B ∩ C) = (A ∩ B) ∩ C

Example (Associative law of union)
Let A = {1,2,3,4}, B = {0,2,4,6} and C = {0,4,5}.

B ∪ C = {0,2,4,5,6}. Then A ∪ (B ∪ C) = {0,1,2,3,4,5,6}.
A ∪ B = {0,1,2,3,4,6}. Then (A ∪ B) ∪ C = {0,1,2,3,4,5,6}.
Both give the same set, so the associative law of union holds.

Example (Associative law of intersection)
Let A = {0,2,4,6}, B = {1,2,3,4} and C = {0,4,5}.

Step 1️⃣: Find B ∩ C = {4}.
Step 2️⃣: Now find A ∩ (B ∩ C) = A ∩ {4} = {4}.

Step 3️⃣: Find A ∩ B = {2,4}.
Step 4️⃣: Then (A ∩ B) ∩ C = {2,4} ∩ {0,4,5} = {4}.

✅ Both results match, so the associative law of intersection is verified.

Remember: When combining three sets, always solve the inner bracket first — especially for intersection — then move outward.

Distributive Laws

The distributive laws relate union and intersection. They are important because they show how one operation distributes over the other:

• A ∪ (B ∩ C) = (A ∪ B) ∩ (A ∪ C) — Union distributes over intersection.
• A ∩ (B ∪ C) = (A ∩ B) ∪ (A ∩ C) — Intersection distributes over union.

Example (Distributive law — union over intersection)
Let A = {1,3,5}, B = {0,5,7}, C = {3,4,5}.

LHS: A ∪ (B ∩ C) = {1,3,5} ∪ ({0,5,7} ∩ {3,4,5}) = {1,3,5} ∪ {5} = {1,3,5}.

RHS: (A ∪ B) ∩ (A ∪ C) = ({1,3,5} ∪ {0,5,7}) ∩ ({1,3,5} ∪ {3,4,5})
= {0,1,3,5,7} ∩ {1,3,4,5} = {1,3,5}.

LHS = RHS, so union distributes over intersection.

Example (Distributive law — intersection over union)
Let A = {3,9,12}, B = {2,6,12} and C = {6,12,18}.

LHS: A ∩ (B ∪ C) = {3,9,12} ∩ ({2,6,12} ∪ {6,12,18}) = {3,9,12} ∩ {2,6,12,18} = {12}.

RHS: (A ∩ B) ∪ (A ∩ C) = ({3,9,12} ∩ {2,6,12}) ∪ ({3,9,12} ∩ {6,12,18})
= {12} ∪ {12} = {12}.

LHS = RHS, so intersection distributes over union.

Exercises — Practice

1. Verify the commutative law of union and intersection for sets A = {1,2,3} and B = {2,3,4}.
2. For A = {0,2,4}, B = {1,2,3}, C = {0,4,5}, verify A ∪ (B ∪ C) = (A ∪ B) ∪ C.
3. For A = {0,2,4,6}, B = {1,2,3,4}, C = {0,4,5}, verify A ∩ (B ∩ C) = (A ∩ B) ∩ C.
4. Verify A ∪ (B ∩ C) = (A ∪ B) ∩ (A ∪ C) for A = {1,3,5}, B = {0,5,7}, C = {3,4,5}.
5. Verify A ∩ (B ∪ C) = (A ∩ B) ∪ (A ∩ C) for A = {3,9,12}, B = {2,6,12}, C = {6,12,18}.

🎯 Operations on Sets — Quick Quiz (Class 8)

Check your understanding of union, intersection, and distributive laws. 🌸

Q1: State the commutative laws of union and intersection.

✅ A ∪ B = B ∪ A and A ∩ B = B ∩ A.

Q2: Write one example to verify the associative law of union.

✅ Example: A = {1,2}, B = {2,3}, C = {3,4}
A ∪ (B ∪ C) = (A ∪ B) ∪ C = {1,2,3,4}

Q3: What do the distributive laws of sets show?

✅ They show how union distributes over intersection and vice versa:
(i) A ∪ (B ∩ C) = (A ∪ B) ∩ (A ∪ C)
(ii) A ∩ (B ∪ C) = (A ∩ B) ∪ (A ∩ C)

Q4: If A = {1,2}, B = {2,3}, find A ∪ B and A ∩ B.

✅ A ∪ B = {1,2,3},   A ∩ B = {2}

Q5: True or False: (A ∪ B)′ = A′ ∪ B′

❌ False — Correct is (A ∪ B)′ = A′ ∩ B′

Try It Yourself:
If A = {a, b} and B = {b, c}, find A ∪ B and A ∩ B.
Answer: A ∪ B = {a, b, c},   A ∩ B = {b}.

🔙 Read Previous Topic: Understanding Sets

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© Math with Raabi — Class 8 Mathematics Notes | Operations on Sets explained with examples.