Fundamentals of Business Mathematics and Statistics · Set Theory, including Venn Diagram

Laws of Sets and De Morgan's Laws Explained

Updated 10 October 2026 · Fact-checked

The laws of sets are rules that let you rewrite expressions with union, intersection and complement, much like algebra rules. They include commutative, associative, distributive, idempotent, identity and De Morgan's laws. To solve a question, apply the right law, or check the identity with a Venn diagram or sample sets.

Understand Laws of Sets and De Morgan's Laws

Sets can be combined with union (∪), intersection (∩) and complement (′). These operations follow fixed rules, called the laws of algebra of sets. They work like the rules of ordinary algebra, but the symbols stand for sets, not numbers.

Some laws will feel familiar. Commutative laws say order does not matter. Associative laws say grouping does not matter when you use only unions or only intersections. Idempotent laws say combining a set with itself gives the same set. Identity laws use the empty set ∅ and the universal set U, which leave a set unchanged or give back ∅ or U.

The distributive laws are different from the others. Union spreads over intersection, and intersection spreads over union. Both directions are true for sets. This is unlike numbers, where addition does not spread over multiplication.

De Morgan's laws tell you what happens when you take the complement of a union or an intersection. The complement of a union is the intersection of the complements. The complement of an intersection is the union of the complements. In words: flip the operation and complement each set.

To prove an identity, show that each side is a subset of the other. Take any element x in the left side and show it lies in the right side, then reverse it. In MCQs you rarely need the full proof. You only need to know the laws well enough to simplify, or to test with small sets.

Key formulas to remember

Commutative laws: A ∪ B = B ∪ A; A ∩ B = B ∩ A; Order of the two sets does not matter.
Associative laws: (A ∪ B) ∪ C = A ∪ (B ∪ C); (A ∩ B) ∩ C = A ∩ (B ∩ C); Works only when the same operation is repeated.
Distributive laws: A ∪ (B ∩ C) = (A ∪ B) ∩ (A ∪ C); A ∩ (B ∪ C) = (A ∩ B) ∪ (A ∩ C); Each operation distributes over the other.
Idempotent laws: A ∪ A = A; A ∩ A = A; A set combined with itself does not change.
Identity laws: A ∪ ∅ = A; A ∩ U = A; A ∪ U = U; A ∩ ∅ = ∅; ∅ is the identity for union, U is the identity for intersection.
Complement laws: A ∪ A′ = U; A ∩ A′ = ∅; (A′)′ = A; ∅′ = U; U′ = ∅; A set and its complement together make up U and share nothing.
De Morgan's laws: (A ∪ B)′ = A′ ∩ B′; (A ∩ B)′ = A′ ∪ B′; Flip ∪ to ∩ (or the reverse) and complement each set.
Difference in terms of complement: A − B = A ∩ B′; Useful for converting differences before applying laws.

How to solve Laws of Sets and De Morgan's Laws questions

Use this method for any question that asks you to prove, simplify or verify a set identity.

1Write the identity and note which sets and operations appear on each side.
2Decide the approach: apply laws, use element-wise proof, or verify with given sets.
3If verifying with sets, first find the universal set U and any complements needed.
4Work out the left side step by step and write its result as a set.
5Work out the right side separately in the same way.
6Compare the two results. If they are equal, the law is verified for those sets.
7For a proof, take a general element x, show LHS ⊆ RHS, then RHS ⊆ LHS, and conclude equality.
8Check that you used the correct law, especially the flip in De Morgan's laws.

Quickest way: Test with small sets and eliminate options

When to use it: Use this when an MCQ asks which identity is true or what an expression simplifies to.

Pick a simple U such as {1, 2, 3, 4}, and small sets A, B, C inside it.
Compute each option's value quickly and compare with the given expression.
Drop any option that fails once. One counter-example is enough.
For De Morgan, remember: complement of union becomes intersection, and the reverse.
For simplification, use A ∪ A′ = U and A ∩ A′ = ∅ to cancel terms early.
Use two different test choices if more than one option survives.

Common mistakes in Laws of Sets and De Morgan's Laws

Writing (A ∪ B)′ = A′ ∪ B′
Students complement each set but forget to flip the operation.
Fix: Say it aloud: complement of union is intersection of complements. The sign always flips.
Treating the distributive law as true in only one direction
Number algebra only allows multiplication over addition.
Fix: For sets, both A ∪ (B ∩ C) and A ∩ (B ∪ C) distribute. Learn both forms.
Computing complement without the universal set
Students take the complement relative to the wrong set or ignore U.
Fix: Always write U first. A′ means elements of U not in A.
Comparing only the first step when verifying a law
Students stop after one side and assume equality.
Fix: Evaluate both sides fully and write both final sets before concluding.
Applying associativity to mixed operations
Students move brackets in A ∪ (B ∩ C) as if both signs were the same.
Fix: Associative laws need the same operation throughout. For mixed signs, use distributive law.
Treating one example as a proof
Verifying with sample sets feels like proving the law.
Fix: An example only verifies the law for those sets. A proof must work for any element x.

Worked examples

Example 1

Let U = {1, 2, 3, 4, 5, 6, 7, 8}, A = {1, 2, 3, 4} and B = {3, 4, 5, 6}. Verify that (A ∪ B)′ = A′ ∩ B′.

Show the solution

A ∪ B = {1, 2, 3, 4, 5, 6}.
(A ∪ B)′ = U − (A ∪ B) = {7, 8}.
A′ = {5, 6, 7, 8} and B′ = {1, 2, 7, 8}.
A′ ∩ B′ = {7, 8}.
Both sides equal {7, 8}.

Answer: (A ∪ B)′ = A′ ∩ B′ = {7, 8}, so De Morgan's law is verified for these sets.

Example 2

Prove that A ∩ (B ∪ C) = (A ∩ B) ∪ (A ∩ C).

Show the solution

Let x ∈ A ∩ (B ∪ C). Then x ∈ A and x ∈ B ∪ C.
So x ∈ A and (x ∈ B or x ∈ C).
If x ∈ B, then x ∈ A ∩ B. If x ∈ C, then x ∈ A ∩ C.
Hence x ∈ (A ∩ B) ∪ (A ∩ C). So LHS ⊆ RHS.
Now let x ∈ (A ∩ B) ∪ (A ∩ C). Then x ∈ A ∩ B or x ∈ A ∩ C.
In both cases x ∈ A, and x is in B or in C, so x ∈ B ∪ C.
Thus x ∈ A ∩ (B ∪ C). So RHS ⊆ LHS.
Each side is a subset of the other.

Answer: A ∩ (B ∪ C) = (A ∩ B) ∪ (A ∩ C), as required.

Exam tips

Memorise De Morgan's laws in words: complement of union is intersection of complements, and the reverse.
In MCQs, test options with a small U and tiny sets instead of proving anything.
Look for A ∪ A′ and A ∩ A′ in an expression. They simplify to U and ∅ at once.
Convert A − B to A ∩ B′ before simplifying a difference expression.
When verifying a law with given sets, write U, each complement and both sides clearly to avoid slips.

Practice questions from Set Theory, including Venn Diagram

Laws of Sets and De Morgan's Laws: frequently asked questions

What are De Morgan's laws in sets?

They state that (A ∪ B)′ = A′ ∩ B′ and (A ∩ B)′ = A′ ∪ B′. The complement of a union is the intersection of the complements. The complement of an intersection is the union of the complements.

How do I prove a distributive law of sets?

Take a general element x in the left side and show it belongs to the right side. Then do the reverse. When each side is a subset of the other, the two sets are equal.

Is a Venn diagram enough to prove a set identity?

A Venn diagram is a good check and is fine for answering an MCQ. A formal proof needs the element-wise method or the laws of sets. Shade both sides and compare the regions.

Is the associative law true for mixed union and intersection?

No. Associativity holds when the same operation is repeated, such as all unions or all intersections. For mixed expressions, use the distributive laws.