Sets – Advanced Concepts: Number Notations, Cardinality, Equal vs Equivalent, Subset, Proper Subset & Superset

After learning the basics of sets, the next portion is advanced set theory — and this is exactly where most JKBOSE board questions and JKSSB MCQs come from.
In exams like JKSSB (FAA, Junior Assistant, Patwari, Police) you usually get 1–3 direct questions from these topics.

This chapter focuses on:

Number set symbols (N, W, Z, Q, R, C)
Cardinality
Equal vs Equivalent sets
Subset & Proper subset
Superset and important properties

1. Number Set Notations (VERY IMPORTANT)

You must memorize these symbols. Almost every exam asks at least one question from here.

Symbol	Name	Meaning	Example
N	Natural Numbers	Counting numbers from 1	{1,2,3,4…}
W	Whole Numbers	Natural numbers + 0	{0,1,2,3…}
Z	Integers	Positive, negative & zero	{…−2,−1,0,1,2…}
Q	Rational Numbers	p/q form, q≠0	1/2, −3/4, 5
R	Real Numbers	Rational + irrational	√2, π, 1.25
C	Complex Numbers	a+bi form	3+4i

Important Relation (Memorize)

N ⊂ W ⊂ Z ⊂ Q ⊂ R ⊂ C

Exam Trap

√2, π → Real but NOT Rational
0 → belongs to W and Z (but not N)

2. Cardinality (Order of a Set)

The cardinal number tells how many elements a set contains.

Notation: n(A) or |A|

Examples:

A = {1,2,3,4} → n(A)=4
B = {2,4,6} → n(B)=3
∅ (empty set) → n(∅)=0
Natural numbers → infinite

Important Rule:
Repeated elements are counted only once.

Example:
A={1,1,2,2,3} → n(A)=3

3. Equal Sets vs Equivalent Sets

This is one of the most asked JKSSB questions.

Equal Sets	Equivalent Sets
Same elements	Same number of elements
A = B	n(A)=n(B)
Elements must match	Elements can be different

Examples:

{1,2,3} and {3,1,2} → Equal
{1,2,3} and {a,b,c} → Equivalent

Exam Trick

Equal ⇒ Always equivalent
Equivalent ⇒ Not always equal

4. Subset (⊆)

A set A is subset of B if every element of A is in B.

Example: A={1,2}
B={1,2,3,4}
Then A ⊆ B

Important Properties

Every set is subset of itself → A ⊆ A
Empty set is subset of every set → ∅ ⊆ A

5. Proper Subset vs Improper Subset

Type	Meaning
Proper subset (A ⊂ B)	A is inside B but not equal
Improper subset	A = B

Example: {1,2} ⊂ {1,2,3} → Proper
{1,2,3} ⊆ {1,2,3} → Improper

6. Superset (⊇)

If A ⊆ B, then B is called superset of A.

Example: A={1,2}
B={1,2,3,4}
Then B ⊇ A

Important Facts to Memorize

A ⊆ A (always true)
∅ ⊆ every set
If A ⊆ B and B ⊆ A → A=B
N ⊂ Z ⊂ Q ⊂ R ⊂ C
Proper subset means smaller set

Quick Revision (FAQs)

Q. Is empty set subset of every set?
Yes.

Q. Is every set a proper subset of itself?
No. Only subset, not proper subset.

Q. Difference between equal and equivalent sets?
Equal → same elements
Equivalent → same number of elements

Q. When A ⊆ B and B ⊆ A?
A=B

JKSSB / Competitive Exam PYQs & MCQs

1. If A={1,2,3} and B={3,2,1}, then
(a) A⊂B
(b) A=B ✔️
(c) A~B only
(d) Not related

2. Cardinality of ∅ is
(a) 1
(b) 0 ✔️
(c) Infinite
(d) Undefined

3. Which number belongs to real but not rational?
(a) 4
(b) 0
(c) √2 ✔️
(d) 3/4

4. If n(A)=n(B), sets are
(a) Equal
(b) Equivalent ✔️
(c) Subset
(d) Disjoint

5. Every set is subset of
(a) Universal set
(b) Empty set
(c) Itself ✔️
(d) None

6. ∅ ⊆ A means
(a) False
(b) True ✔️
(c) Undefined
(d) Only for finite sets

7. {1,2} ⊆ {1,2,3} represents
(a) Proper subset ✔️
(b) Improper subset
(c) Equal sets
(d) Equivalent

8. Which is correct chain?
(a) Z ⊂ N
(b) R ⊂ Q
(c) N ⊂ Z ⊂ Q ⊂ R ✔️
(d) Q ⊂ N

9. If A⊆B and B⊆A then
(a) A⊂B
(b) B⊂A
(c) A=B ✔️
(d) None

10. n({1,1,2,2,3}) = ?
(a) 5
(b) 4
(c) 3 ✔️
(d) 2

Exam Tip

From Sets chapter, examiners mainly ask:

Symbols (N, Z, Q, R)
Empty set property
Equal vs Equivalent
Subset questions

Prepare this chapter properly → Guaranteed 2–4 marks in JKSSB.

Next topic: Operations on Sets (Union, Intersection & Venn Diagrams) — also very important for exams.