De Morgan’s Laws for Class 8 – Complement of Union and Intersection with Solved Examples | Math With Raabi

 

🧭 De Morgan’s Laws — Complement of Union and Intersection Explained with Examples (Class 8)

Hey learners! 👋 Ready for another fun math journey? In the last topic, we learned about Union and Intersection of sets. Today, we explore De Morgan’s Laws with step-by-step solved examples! 🌟 These laws connect Union, Intersection, and Complement, making complex set problems easier. ✨

Before you start: Check these earlier posts first:

• Operations on Sets – Union & Intersection
• Recognition of Important Sets & Subsets
• Understanding Sets in Mathematics (Class 8)

📜 A Quick History of De Morgan’s Laws

Named after Augustus De Morgan (1806–1871), these laws reveal the relationship between Union, Intersection, and Complement. They are fundamental in logic, computer science, and mathematics. 💡

🌼 Understanding the Complement of a Set

If U is the universal set and A is a subset of U, then the complement of A (A′) contains all elements of U not in A.

A′ = { x ∈ U : x ∉ A }

💭 Remember: Complement means “everything outside the set!”

🔹 Statement of De Morgan’s Laws

• (A ∪ B)′ = A′ ∩ B′ — complement of a union equals intersection of complements
• (A ∩ B)′ = A′ ∪ B′ — complement of an intersection equals union of complements

🧠 Think: NOT (A OR B) and NOT (A AND B)

Example 1 — Verify (A ∪ B)′ = A′ ∩ B′

Let U = {a, b, c, d, e, f}, A = {b, d, f}, B = {a, b, e, f}.

LHS: A ∪ B = {a, b, d, e, f}, (A ∪ B)′ = {c}.

RHS: A′ = {a, c, e}, B′ = {c, d}, A′ ∩ B′ = {c}.

✅ Verified: (A ∪ B)′ = A′ ∩ B′ = {c}

Example 2 — Verify (A ∩ B)′ = A′ ∪ B′

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

LHS: A ∩ B = {2}, (A ∩ B)′ = {1, 3, 4, 5, 6}.

RHS: A′ = {3, 5, 6}, B′ = {1, 4, 5}, A′ ∪ B′ = {1, 3, 4, 5, 6}.

✅ Verified: (A ∩ B)′ = A′ ∪ B′ = {1, 3, 4, 5, 6}

🧮 Practice Problems

1. For U = {a,b,c,d,e}, A = {a,d}, B = {b,d,e}, verify both De Morgan’s Laws.
2. For U = {1–6}, A = {1,2,3}, B = {2,4,6}, compare (A ∪ B)′ with A′ ∩ B′.
3. Explain why complement changes union ↔ intersection in your own words.

🎯 De Morgan’s Laws — Quick Quiz (Class 8)

Test your understanding of union, intersection, and complements. 🌸

Q1: Write the first De Morgan’s Law in symbolic form.

✅ (A ∪ B)′ = A′ ∩ B′

Q2: Write the second De Morgan’s Law in symbolic form.

✅ (A ∩ B)′ = A′ ∪ B′

Q3: What does the complement of a set represent?

✅ All elements of the universal set not in the given set.

Q4: If U = {1, 2, 3, 4, 5} and A = {1, 3, 5}, find A′.

✅ A′ = {2, 4}

Q5: How does taking the complement affect union and intersection?

✅ Complement changes Union ↔ Intersection.

🔙 Read Previous Topic: Operations on Sets

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