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.