Set Operations and Laws of Algebra of Sets | NIMCET

Quick answer: The main set operations are union $A \cup B$ (elements in A or B), intersection $A \cap B$ (in both), difference $A - B$ (in A but not B), complement $A'$ (in U but not A) and symmetric difference $A \Delta B = (A - B) \cup (B - A)$ . The most tested laws are De Morgan's laws: $(A \cup B)' = A' \cap B'$ and $(A \cap B)' = A' \cup B'$ .

This article builds on sets, subsets and power sets.

The operations

Operation	Notation	Definition
Union	$A \cup B$	$\{x : x \in A \text{ or } x \in B\}$
Intersection	$A \cap B$	$\{x : x \in A \text{ and } x \in B\}$
Difference	$A - B$	$\{x : x \in A, x \notin B\}$
Complement	$A'$	$\{x \in U : x \notin A\}$
Symmetric difference	$A \Delta B$	$(A - B) \cup (B - A)$

Two sets are disjoint if $A \cap B = \phi$ .

A quick example

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

Expression	Result
$A \cup B$	$\{1, 2, 3, 4, 5, 6\}$
$A \cap B$	$\{3, 4\}$
$A - B$	$\{1, 2\}$
$B - A$	$\{5, 6\}$
$A \Delta B$	$\{1, 2, 5, 6\}$
$A'$	$\{5, 6, 7, 8\}$

Laws of the algebra of sets

Law	Union form	Intersection form
Idempotent	$A \cup A = A$	$A \cap A = A$
Identity	$A \cup \phi = A$	$A \cap U = A$
Domination	$A \cup U = U$	$A \cap \phi = \phi$
Commutative	$A \cup B = B \cup A$	$A \cap B = B \cap A$
Associative	$(A \cup B) \cup C = A \cup (B \cup C)$	$(A \cap B) \cap C = A \cap (B \cap C)$
Distributive	$A \cup (B \cap C) = (A \cup B) \cap (A \cup C)$	$A \cap (B \cup C) = (A \cap B) \cup (A \cap C)$
Complement	$A \cup A' = U$	$A \cap A' = \phi$
De Morgan	$(A \cup B)' = A' \cap B'$	$(A \cap B)' = A' \cup B'$
Absorption	$A \cup (A \cap B) = A$	$A \cap (A \cup B) = A$

Also: $(A')' = A$ , $U' = \phi$ and $\phi' = U$ .

Useful identities for difference

Identity
$A - B = A \cap B'$
$A - (B \cup C) = (A - B) \cap (A - C)$
$A - (B \cap C) = (A - B) \cup (A - C)$
$A \Delta B = (A \cup B) - (A \cap B)$

The first identity is the most useful: it turns every difference into an intersection, so the other laws apply directly.

Properties of symmetric difference

Commutative: $A \Delta B = B \Delta A$ .
Associative: $(A \Delta B) \Delta C = A \Delta (B \Delta C)$ .
$A \Delta \phi = A$ and $A \Delta A = \phi$ .
$n(A \Delta B) = n(A) + n(B) - 2n(A \cap B)$ .

Worked examples

Example 1: simplify using laws

Simplify $(A \cup B) \cap (A \cup B')$ .

By the distributive law, this is $A \cup (B \cap B') = A \cup \phi = A$ .

Example 2: prove a difference identity

Show that $A - (A - B) = A \cap B$ .

Step 1: $A - (A - B) = A \cap (A \cap B')'$ .

Step 2: By De Morgan, $(A \cap B')' = A' \cup B$ .

Step 3: $A \cap (A' \cup B) = (A \cap A') \cup (A \cap B) = \phi \cup (A \cap B) = A \cap B$ .

Example 3: symmetric difference count

If $n(A) = 12$ , $n(B) = 9$ and $n(A \cap B) = 4$ , find $n(A \Delta B)$ .

$12 + 9 - 2 \cdot 4 = 13$ .

Example 4: when is A ∪ B = A ∩ B?

This happens only when $A = B$ . If some element were in A but not B, it would be in the union but not the intersection.

Summary table

Idea	Key result
Difference	$A - B = A \cap B'$
De Morgan	Complement swaps ∪ and ∩
Symmetric difference	$(A \cup B) - (A \cap B)$
Count of $A \Delta B$	$n(A) + n(B) - 2n(A \cap B)$
$A \cup B = A \cap B$	Only if $A = B$

Exam tips

Replace $A - B$ by $A \cap B'$ before simplifying.
For identity questions, test with small sets or a Venn diagram to rule out wrong options quickly.
Learn De Morgan's laws in words: "the complement of a union is the intersection of complements".

Next, see Venn diagrams and the inclusion-exclusion principle or the NIMCET Set Theory chapter.

Common mistakes

Treating $A - B$ and $B - A$ as the same set.
Writing $(A \cup B)' = A' \cup B'$ ; the operation must switch.
Forgetting that the complement depends on the universal set.
Confusing disjoint sets with equal-size sets.

Frequently Asked Questions

What is De Morgan's law in sets?

(A ∪ B)′ = A′ ∩ B′ and (A ∩ B)′ = A′ ∪ B′. Taking a complement switches union and intersection.

What is the symmetric difference of two sets?

A Δ B is the set of elements in exactly one of A and B. It equals (A − B) ∪ (B − A).

How is A − B written using intersection?

A − B = A ∩ B′.

What does it mean for two sets to be disjoint?

Two sets are disjoint if they have no common element, that is, A ∩ B = φ.

What is the distributive law for sets?

A ∩ (B ∪ C) = (A ∩ B) ∪ (A ∩ C), and A ∪ (B ∩ C) = (A ∪ B) ∩ (A ∪ C).