Class 11 | CBSE | MATHEMATICS
MATHEMATICS | CONCEPT OF SUB SET

Big Idea (Anchor Thought)

A subset is a smaller group living completely inside a bigger group.

Visualise boxes inside boxes 📦. Your brain remembers visuals faster than definitions.

Definition (In Simple Words)

A set A is called a subset of a set B if every element of A is also present in B.

We write this as:

\[ A \subset B \]

Brain Shortcut (One-Line Rule)

If something is in A, it must be in B.

Mathematically:

\[ A \subset B \quad \text{if} \quad a \in A \Rightarrow a \in B \]

Meaning of the Symbol “⇒”

The symbol ⇒ means implies.

\[ a \in A \Rightarrow a \in B \]

This means: If an element belongs to A, then it must belong to B.

When A is NOT a Subset of B

If even one element of A is not present in B, then A is not a subset of B.

We write:

\[ A \nsubseteq B \]

Brain trigger: One wrong element breaks the entire rule.

Can B Be a Subset of A and A be a Subset of B?

Yes, sometimes.

If every element of A is in B and every element of B is in A, then both sets are exactly the same.

\[ A \subset B \;\; \text{and} \;\; B \subset A \;\; \Leftrightarrow \;\; A = B \]

Meaning of the Symbol “⇔”

The symbol ⇔ means “if and only if” (iff).

Both directions must be true — forward ✔ and backward ✔.

⭐ Important Results (Exam Gold)

✅ Every set is a subset of itself

\[ A \subset A \]

✅ Empty set is a subset of every set

\[ \varnothing \subset A \]

Since the empty set has no elements, there is nothing to violate the subset rule.

Final Brain Snapshot

• Subset means fully inside
• One wrong element means not a subset
• Two-way subset means equal sets
• Every set is a subset of itself
• Empty set is a subset of every set

Given Sets

φ = Empty set
A = { 1, 3 }
B = { 1, 5, 9 }
C = { 1, 3, 5, 7, 9 }

We have to insert either ⊂ (subset) or ⊄ (not a subset).

(i) φ ⊂ B

Answer: φ ⊂ B

Brain Logic:
The empty set has no elements. Since there is nothing inside φ, there is also nothing that can violate the subset rule.

✔ Therefore, the empty set is a subset of every set.

(ii) A ⊄ B

Answer: A ⊄ B

Check element by element:
A = { 1, 3 }
B = { 1, 5, 9 }

✔ 1 is in B
❌ 3 is not in B

Neural Trigger: Just one missing element is enough to break the subset condition.

✔ Hence, A ⊄ B

(iii) A ⊂ C

Answer: A ⊂ C

Compare elements:
A = { 1, 3 }
C = { 1, 3, 5, 7, 9 }

✔ 1 is in C
✔ 3 is in C

Brain Pattern:
All elements of A are found inside C. Extra elements in C do not matter.

✔ Therefore, A ⊂ C

(iv) B ⊂ C

Answer: B ⊂ C

Element check:
B = { 1, 5, 9 }
C = { 1, 3, 5, 7, 9 }

✔ 1 is in C
✔ 5 is in C
✔ 9 is in C

Memory Hook:
When every element passes the check, the our brain confidently accepts “subset”.

✔ Hence, B ⊂ C

Final Summary (Very Important)

Given Statement

Let A, B, C be three sets.

If A ∈ B and B ⊂ C, is it true that A ⊂ C?

If not, give an example.

Brain Alert (Key Confusion Area)

Students often confuse:

• ∈ → means element of
• ⊂ → means subset of

Your brain must treat them as completely different ideas.

❓ Is the Statement True?

Answer: ❌ No, it is not true.

📌 Counter Example (Very Important)

Let:

A = {1}
B = {{1}, 2}
C = {{1}, 2, 3}

Step-by-Step Brain Verification

Step 1: Check A ∈ B

A = {1}
B contains {1} as an element.

✔ Therefore, A ∈ B

Step 2: Check B ⊂ C

All elements of B = {{1}, 2} are present in C = {{1}, 2, 3}.

✔ Therefore, B ⊂ C

Step 3: Check A ⊂ C

A = {1} → element inside A is 1

But 1 ∉ C (C contains {1}, not 1).

❌ Therefore, A ⊄ C

Why the Brain Gets Confused Here

The brain automatically assumes:

If something is inside B and B is inside C, then it must be inside C.

❌ This logic works for subsets, but not for elements.

⭐ Extremely Important Note (Exam Favourite)

An element of a set can never be a subset of itself.

Example:

1 ∈ {1} but 1 ⊄ {1}

Because subsets talk about collections, not single elements.

Final Brain Takeaway

Big Brain Idea (Anchor Thought)

The set of real numbers R is like a big universe. Inside it live many smaller but very important sets.

The brain learns faster when it sees hierarchy — smaller sets inside bigger sets.

Set of Natural Numbers (N)

The set of natural numbers is:

\[ N = \{1, 2, 3, 4, 5, \ldots\} \]

Brain Cue:
These are the numbers we naturally use for counting.

Set of Integers (Z)

The set of integers is:

\[ Z = \{\ldots, -3, -2, -1, 0, 1, 2, 3, \ldots\} \]

Brain Expansion:
Integers include negative numbers and zero, so they are bigger than natural numbers.

Set of Rational Numbers (Q)

The set of rational numbers is defined as:

\[ Q = \{ x : x = \frac{p}{q}, \; p, q \in Z \; \text{and} \; q \neq 0 \} \]

This is read as: Q is the set of all numbers x such that x can be written as p/q, where p and q are integers and q is not zero.

Memory Hook:
If a number can be written as a fraction, it is rational.

Examples:
−5 = −5/1
5/7
3 + 1/2

Set of Irrational Numbers (T)

The set of irrational numbers is denoted by T.

\[ T = \{ x : x \in R \; \text{and} \; x \notin Q \} \]

Brain Logic:
Irrational numbers are real numbers that cannot be written as fractions.

Examples:
√2 , √5 , π

Important Relations Among These Sets

Some obvious and very important subset relations are:

\[ N \subset Z \subset Q \]

\[ Q \subset R \]

\[ T \subset R \]

\[ N \nsubseteq T \]

Brain Reminder:
Natural numbers are rational, but they are not irrational.

Final Brain Snapshot (One-Glance Memory)

• Natural numbers ⊂ Integers ⊂ Rational numbers
• Rational and irrational numbers together form real numbers
• Irrational numbers cannot be written as p/q
• Natural numbers are not a subset of irrational numbers