Q.Make correct statements by filling in the symbols ⊂ or ⊄ in the blank spaces :

Concept: Set Membership — we check whether every element of the first set is also present in the second set. If yes, use ⊂; otherwise, use ⊄.

1. {2, 3, 4} ⊂ {1, 2, 3, 4, 5} — all three elements are in the second set.
2. {a, b, c} ⊄ {b, c, d} — a is missing from the second set.
3. {Class XI students} ⊂ {all students of your school} — every Class XI student is a student of the school.
4. {all circles} ⊄ {circles of radius 1} — a circle of radius 2, for example, is not in the second set.
5. {triangles} ⊄ {rectangles} — no triangle is a rectangle.
6. {equilateral triangles} ⊂ {all triangles} — every equilateral triangle is a triangle.
7. {even natural numbers} ⊂ {integers} — every even natural number is an integer.
   ✓Final answer

   1. ⊂,
   2. ⊄,
   3. ⊂,
   4. ⊄,
   5. ⊄,
   6. ⊂,
   7. ⊂

Concept first: Set membership and subsets

A set $A$ is a subset of $B$ (written $A \subset B$) if every element of $A$ is also an element of $B$. If even one element of $A$ is missing from $B$, then $A \not\subset B$. This is a pure "all-or-nothing" check — no partial credit in set theory.

The trick is to look at the elements, not the labels. For sets described by conditions, translate the condition into actual elements before comparing.

Let's go through each part.

1. $\{2, 3, 4\} \dots \{1, 2, 3, 4, 5\}$

   Every element of the first set — 2, 3, and 4 — appears in the second set. So the first is a subset of the second.

   $\boxed{\subset}$

2. $\{a, b, c\} \dots \{b, c, d\}$

   The first set contains $a$, but $a$ is not in $\{b, c, d\}$. Since one element fails, it is not a subset.

   $\boxed{\not\subset}$

3. $\{x : x \text{ is a student of Class XI of your school}\} \dots \{x : x \text{ is a student of your school}\}$

   Every Class XI student is, by definition, a student of the school. So the first set is entirely contained in the second.

   $\boxed{\subset}$

4. $\{x : x \text{ is a circle in the plane}\} \dots \{x : x \text{ is a circle in the same plane with radius 1 unit}\}$

   The first set includes circles of any radius — radius 2, radius 5, etc. The second set only contains circles of radius exactly 1. So a circle of radius 2 is in the first set but not in the second.

   $\boxed{\not\subset}$

   Watch out

   A common mistake is to reverse the direction: the set of radius‑1 circles is a subset of all circles, not the other way around. Here we are checking if all circles are radius‑1 circles — which is false.

5. $\{x : x \text{ is a triangle in a plane}\} \dots \{x : x \text{ is a rectangle in the plane}\}$

   A triangle is never a rectangle, and a rectangle is never a triangle. The two sets have no overlap at all. So the first set is certainly not a subset of the second.

   $\boxed{\not\subset}$

6. $\{x : x \text{ is an equilateral triangle in a plane}\} \dots \{x : x \text{ is a triangle in the same plane}\}$

   Every equilateral triangle is a triangle. So the first set is fully contained in the second.

   $\boxed{\subset}$

7. $\{x : x \text{ is an even natural number}\} \dots \{x : x \text{ is an integer}\}$

   Even natural numbers (2, 4, 6, …) are all integers. So the first set is a subset of the second.

   $\boxed{\subset}$

Method: Definition of Subset (⊂) and Not a Subset (⊄)

Concept:

A set $A$ is a subset of set $B$ (written $A \subset B$) if every element of $A$ is also an element of $B$.

If there is at least one element in $A$ that is not in $B$, then $A \not\subset B$.

Steps to decide ⊂ or ⊄:

1. List or describe the elements of both sets clearly.
2. Check each element of the first set — is it present in the second set?
3. If all are present → use ⊂.
4. If any one is missing → use ⊄.

Applying the method:

(i) $\{2, 3, 4\} \dots \{1, 2, 3, 4, 5\}$

• Every element (2, 3, 4) is in the second set.
• Answer: $\{2, 3, 4\} \subset \{1, 2, 3, 4, 5\}$

(ii) $\{a, b, c\} \dots \{b, c, d\}$

• $a$ is not in $\{b, c, d\}$.
• Answer: $\{a, b, c\} \not\subset \{b, c, d\}$

(iii) $\{x : x \text{ is a student of Class XI of your school}\} \dots \{x : x \text{ student of your school}\}$

• Every Class XI student is a student of the school.
• Answer: $\subset$

(iv) $\{x : x \text{ is a circle in the plane}\} \dots \{x : x \text{ is a circle in the same plane with radius 1 unit}\}$

• A circle of radius 2 is in the first set but not in the second.
• Answer: $\not\subset$

(v) $\{x : x \text{ is a triangle in a plane}\} \dots \{x : x \text{ is a rectangle in the plane}\}$

• A triangle is never a rectangle.
• Answer: $\not\subset$

(vi) $\{x : x \text{ is an equilateral triangle in a plane}\} \dots \{x : x \text{ is a triangle in the same plane}\}$

• Every equilateral triangle is a triangle.
• Answer: $\subset$

(vii) $\{x : x \text{ is an even natural number}\} \dots \{x : x \text{ is an integer}\}$

• Every even natural number (2, 4, 6, …) is an integer.
• Answer: $\subset$

Final filled statements:

1. $\{2, 3, 4\} \subset \{1, 2, 3, 4, 5\}$
2. $\{a, b, c\} \not\subset \{b, c, d\}$
3. $\subset$
4. $\not\subset$
5. $\not\subset$
6. $\subset$
7. $\subset$

Here’s a breakdown of the common mistakes students make with Set Membership and Subset (⊂ / ⊄) questions, using your examples.

🧠 Core Concept Reminder

• ⊂ (subset): Every element of the first set must also be an element of the second set.
• ⊄ (not a subset): At least one element of the first set is not in the second set.

✗ Common Mistake #1: Confusing “element of” with “subset of”

Example from (i):

{ 2, 3, 4 } . . . { 1, 2, 3, 4, 5 }

Mistake: Students write ∈ instead of ⊂.

They think: “2, 3, 4 are in the second set, so it’s a subset.” But they use the wrong symbol.

How to avoid:

• ∈ is for single elements (e.g., 2 ∈ {1,2,3}).
• ⊂ is for whole sets (e.g., {2,3} ⊂ {1,2,3}).
• Check: Is the left side a set (with curly braces)? Then use ⊂ or ⊄.

✓ Correct: { 2, 3, 4 } ⊂ { 1, 2, 3, 4, 5 }

✗ Common Mistake #2: Forgetting to check every element

Example from (ii):

{ a, b, c } . . . { b, c, d }

Mistake: Student sees b and c are in the second set, but forgets a is missing. They write ⊂.

How to avoid:

• Scan each element of the first set.
• Ask: “Is this element in the second set?”
• If even one is missing → ⊄.

✓ Correct: { a, b, c } ⊄ { b, c, d }

✗ Common Mistake #3: Misreading the description (especially “your school”)

Example from (iii):

{x : x is a student of Class XI of your school} . . . {x : x student of your school}

Mistake: Student thinks “Class XI” is the whole school, or ignores “your school” and treats it as any school.

How to avoid:

• Read the condition literally.
• Class XI students are a subset of all students in the same school.
• Every Class XI student is also a student of the school → subset.

✓ Correct: ⊂

✗ Common Mistake #4: Reversing the subset direction

Example from (iv):

{x : x is a circle in the plane} . . . {x : x is a circle in the same plane with radius 1 unit}

Mistake: Student writes ⊂ thinking “circles with radius 1 are a subset of all circles” — which is true — but they put the larger set on the left.

How to avoid:

• Identify which set is bigger.
• All circles include circles of radius 1, but not vice versa.
• So the left set (all circles) is not a subset of the right set (only radius 1).

✓ Correct: ⊄

✗ Common Mistake #5: Assuming different shapes can be subsets

Example from (v):

{x : x is a triangle in a plane} . . . {x : x is a rectangle in the plane}

Mistake: Student thinks “triangle” and “rectangle” are both shapes, so maybe one is a subset. They write ⊂.

How to avoid:

• No triangle is a rectangle.
• The two sets have no common elements.
• If they share zero elements, the first set cannot be a subset of the second.

✓ Correct: ⊄

✗ Common Mistake #6: Ignoring “special case” logic

Example from (vi):

{x : x is an equilateral triangle in a plane} . . . {x : x is a triangle in the same plane}

Mistake: Student writes ⊄ because they think “equilateral” is different from “triangle”.

How to avoid:

• Every equilateral triangle is a triangle.
• The first set is a special case of the second.
• Special case → subset.

✓ Correct: ⊂

✗ Common Mistake #7: Confusing “natural numbers” with “integers”

Example from (vii):

{x : x is an even natural number} . . . { x : x is an integer}

Mistake: Student thinks “natural numbers” and “integers” are unrelated, or writes ⊄ because integers include negatives.

How to avoid:

• Natural numbers (usually 1,2,3,…) are a subset of integers (…,-2,-1,0,1,2,…).
• Even natural numbers are still natural numbers → they are integers.
• So every element of the first set is in the second.

✓ Correct: ⊂

📝 Quick Summary Table

Final tip: Always ask yourself — “Is every item from the first set guaranteed to be in the second?” If yes → ⊂. If no → ⊄.