set maths

We have to find:

---

🔹 Step 1: Use the basic formula

P(A \cup B) = P(A) + P(B) - P(A \cap B)

---

🔹 Step 2: Substitute the given values

0.6 = 0.35 + P(B) - 0.25

---

🔹 Step 3: Simplify

0.6 = 0.10 + P(B)

P(B) = 0.6 - 0.10 = 0.5

---

✅ Answer 1:

\boxed{P(B) = 0.5}

---

🔹 Step 4: Find P(not B)

P(\text{not }B) = 1 - P(B)

P(B') = 1 - 0.5 = 0.5

---

✅ Answer 2:

\boxed{P(\text{not }B) = 0.5}

---

🌟 Final Answers

• P(B) = 0.5
• P(not B) = 0.5

---

🧠 Easy Memory Tip

• Union formula → add – subtract
• Not B → 1 − B

Below are the answers with very simple explanations, written in Plus One exam style 👍
(1 mark each – easy to remember)

---

(i) If A ⊂ B, then A ∪ B = ?

✅ Answer:

A \cup B = B

Explanation (Simple):

• A is a subset of B
• All elements of A are already in B
• Union means all elements

👉 So, union gives the bigger set B

---

(ii) Write the set { x : x is a positive integer and x² < 40 } in roster form

Step 1: Check values

• ✔
• ✔
• ✔
• ✔
• ✔
• ✔
• ❌ (stop)

✅ Answer (Roster Form):

\{1, 2, 3, 4, 5, 6\}

Explanation:

Only positive integers whose square is less than 40 are taken.

---

(iii) Write all the subsets of {2}

Rule:

A set with one element has two subsets

✅ Answer:

\varnothing,\; \{2\}

Explanation:

• One subset with no elements → empty set
• One subset with the element itself

---

🌟 Quick Exam Tip

• Subset question → count =
• Roster form → list values clearly
• A ⊂ B → union = B

If you want:

• ✔️ More 1-mark questions
• ✔️ MCQs
• ✔️ Malayalam explanations
• ✔️ One-page revision sheet

Just tell me 😊

Plus One Mathematics

Chapter 1 – SETS (One Page Revision Notes)

---

1. Set

A set is a well-defined collection of distinct objects.

Example: A = {1, 2, 3} Each object is called an element.

---

2. Representation of Sets

• Roster Form: Listing elements
  • A = {2, 4, 6}
• Set Builder Form: Using a property
  • A = {x : x is an even natural number, x < 10}

---

3. Types of Sets

• Empty (Null) Set (∅): No elements
• Singleton Set: Only one element
• Finite Set: Limited elements
• Infinite Set: Unlimited elements
• Equal Sets: Same elements
• Subset: A ⊆ B (all elements of A are in B)

---

4. Universal Set (U)

The set containing all elements under discussion.

---

5. Venn Diagram

A pictorial representation of sets using circles inside a rectangle (U). Used to show relationships between sets.

---

6. Operations on Sets

Let A and B be two sets:

• Union (A ∪ B): Elements in A or B or both
• Intersection (A ∩ B): Common elements of A and B
• Difference (A − B): Elements in A but not in B
• Complement (A′): Elements in U but not in A

---

7. Laws of Sets

• A ∪ ∅ = A
• A ∩ ∅ = ∅
• A ∪ A = A
• A ∩ A = A
• A ∪ A′ = U
• A ∩ A′ = ∅
• (A′)′ = A

---

8. Important Points

• Order of elements does not matter in a set
• Repetition of elements is not allowed
• Sets form the foundation for Relations, Functions, Probability, and Statistics

---

✨ Exam Tips

• Learn definitions exactly
• Practice Venn diagram problems
• Remember symbols: ∪, ∩, ⊆, ∅, U, ′

---

✅ 

• Definitions
• Types of sets
• Operations on sets
• Laws of sets
• Symbols & exam tips

Sure 🙂
Below are the answers with simple explanations for all the basic Set problems.

---

1️⃣ Identify the Set (Roster Form)

a) First 5 natural numbers
👉 Natural numbers start from 1
Answer: {1, 2, 3, 4, 5}

b) Vowels in English alphabet
👉 Vowels are fixed letters
Answer: {a, e, i, o, u}

c) Even numbers less than 10
👉 Even numbers divisible by 2
Answer: {2, 4, 6, 8}

---

2️⃣ True or False

a) {1, 2, 3} = {3, 2, 1}
✔ True
👉 Order does not matter in sets.

b) 0 ∈ {1, 2, 3}
❌ False
👉 0 is not an element of the set.

c) ∅ is a subset of every set
✔ True
👉 Empty set has no elements, so it fits in all sets.

---

3️⃣ Types of Sets

a) {Sunday}
✔ Singleton set
👉 Only one element.

b) {x : x is a negative natural number}
✔ Empty set (∅)
👉 Natural numbers are never negative.

c) {1, 3, 5, 7, 9}
✔ Finite set
👉 Limited number of elements.

d) N = {1, 2, 3, …}
✔ Infinite set
👉 Endless elements.

---

4️⃣ Subset Problems

Given:
A = {1, 2}
B = {1, 2, 3, 4}

a) Is A ⊆ B?
✔ Yes
👉 Every element of A is in B.

b) Is B ⊆ A?
❌ No
👉 3 and 4 are not in A.

---

5️⃣ Operations on Sets

A = {1, 2, 3}
B = {3, 4, 5}

a) A ∪ B (Union = all elements)
👉 {1, 2, 3, 4, 5}

b) A ∩ B (Common elements)
👉 {3}

c) A − B (In A but not in B)
👉 {1, 2}

---

6️⃣ Complement

U = {1, 2, 3, 4, 5, 6}
A = {2, 4, 6}

a) A′
👉 Elements in U but not in A
Answer: {1, 3, 5}

---

7️⃣ One-Mark Questions

a) What is an empty set?
👉 A set with no elements.

b) Symbol for union
👉 ∪

c) Symbol for complement
👉 ′

---

8️⃣ Law-Based Questions

Take A = {1, 2}

a) A ∪ ∅ = A
👉 {1, 2} ∪ ∅ = {1, 2} ✔

b) A ∩ A = A
👉 {1, 2} ∩ {1, 2} = {1, 2} ✔

---

🌟 Quick Exam Tip

• Always write sets in brackets { }
• Use correct symbols
• Explain in one simple line if needed

Sure 😊
I’ll teach the theory first, then give step-by-step solutions for each question shown in your images.
(Exactly at Plus One level, simple language.)

---

🔷 BASIC THEORY (Before Solving)

1️⃣ Subset

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

• Empty set (∅) is a subset of every set.
• A set is also a subset of itself.

If a set has n elements, number of subsets = 2ⁿ

---

2️⃣ Interval Notation

• (a, b) → a < x < b
• [a, b] → a ≤ x ≤ b
• (a, b] → a < x ≤ b
• [a, b) → a ≤ x < b

---

3️⃣ Operations on Sets

• Union (A ∪ B) → all elements of A or B
• Intersection (A ∩ B) → common elements
• Difference (A − B) → elements in A but not in B
• Complement (A′) → elements in U but not in A

---

4️⃣ Important Laws

• If A ⊂ B → A ∪ B = B
• U ∩ A = A
• De Morgan’s Laws
  • (A ∪ B)′ = A′ ∩ B′
  • (A ∩ B)′ = A′ ∪ B′

---

🔷 QUESTION 2

Write all the subsets of A = {1, 2, 3}

Number of elements = 3
Number of subsets = 2³ = 8

Subsets:

1. ∅
2. {1}
3. {2}
4. {3}
5. {1, 2}
6. {1, 3}
7. {2, 3}
8. {1, 2, 3}

✅ Answer complete

---

🔷 QUESTION 3 (Choose the correct answer)

(a)

Given: { x : x ∈ R, 3 < x ≤ 5 }

• 3 is not included
• 5 is included

Correct interval = (3, 5]

✅ Answer: (iv) (3, 5]

---

(b)

If A ⊂ B, then A ∪ B = ?

Union of a smaller set with a bigger set is the bigger set.

✅ Answer: (ii) B

---

(c)

If U is the universal set, then U ∩ A = ?

Intersection with U gives the same set.

✅ Answer: (ii) A

---

🔷 QUESTION 4

Given:
A = {2, 3, 4, 5}
B = {4, 5, 6, 7}

---

(a) A ∪ B

All elements (no repetition):

A ∪ B = {2, 3, 4, 5, 6, 7}

---

(b) A ∩ B

Common elements:

A ∩ B = {4, 5}

---

(c) A − B

Elements in A but not in B:

A − B = {2, 3}

---

(d) B − A

Elements in B but not in A:

B − A = {6, 7}

---

(e) Verify

(A ∪ B) − (A ∩ B) = (A − B) ∪ (B − A)

LHS:
(A ∪ B) = {2,3,4,5,6,7}
(A ∩ B) = {4,5}

LHS = {2,3,6,7}

RHS:
(A − B) ∪ (B − A)
= {2,3} ∪ {6,7}
= {2,3,6,7}

✅ LHS = RHS (Verified)

---

🔷 QUESTION 5

Given:
U = {1,2,3,4,5,6,7,8,9}
A = {2,4,6,8}
B = {2,3,5,7}

---

(a) A′ and B′

A′ = U − A = {1,3,5,7,9}
B′ = U − B = {1,4,6,8,9}

---

(b) A ∪ B

A ∪ B = {2,3,4,5,6,7,8}

---

(c) A ∩ B

Common element = {2}

---

(d) Verify (A ∪ B)′ = A′ ∩ B′

(A ∪ B)′ = {1,9}
A′ ∩ B′ = {1,9}

✅ Verified

---

(e) Verify (A ∩ B)′ = A′ ∪ B′

(A ∩ B) = {2}
(A ∩ B)′ = {1,3,4,5,6,7,8,9}

A′ ∪ B′ = {1,3,4,5,6,7,8,9}

✅ Verified

---

🌟 FINAL EXAM TIPS

• Always write sets in curly brackets { }
• Show steps for verify questions
• Learn De Morgan’s laws perfectly

If you want next: ✔️ More exam problems ✔️ MCQs ✔️ One-page formula sheet ✔️ Venn diagram explanation

Just tell me 😊

Here is a clear theory + explanation, written exactly in exam-friendly language 👌

---

🔷 Given

A ⊂ B

This means:

• A is a subset of B
• Every element of A is already present in B

---

🔷 Theory: Union of Sets

The union of two sets A ∪ B is defined as:

The set of all elements which are in A or in B or in both.

Symbolically:

A \cup B = \{x : x \in A \text{ or } x \in B\}

---

🔷 Key Idea (Very Important)

If A ⊂ B, then:

• A has no extra elements
• B already contains all elements of A

So when we take the union:

• Nothing new is added from A
• The result is simply B

---

🔷 Simple Example (For Understanding)

Let:

• A = {1, 2}
• B = {1, 2, 3, 4}

Here:

• A ⊂ B

Now:

A \cup B = \{1, 2, 3, 4\} = B

---

🔷 Final Result

\boxed{A \cup B = B}

---

✅ Correct Answer

(ii) B

---

✨ Exam Tip (Remember This Rule)

Union of a subset with its superset is the superset itself.

This theory + explanation is perfect for 2-mark questions or MCQs.

If you want:

• More law-based explanations
• MCQs with theory
• Short exam answers

Just tell me 😊

Here is a clear theory + explanation, written in simple exam language 👌

---

🔷 Given

• U is the Universal set
• A is any set

We are asked to find:

U \cap A = \; ?

---

🔷 Theory: Universal Set & Intersection

Universal Set (U)

The universal set contains all elements under consideration.

Intersection of Sets

The intersection of two sets is defined as:

The set of common elements present in both sets.

Symbolically:

A \cap B = \{x : x \in A \text{ and } x \in B\}

---

🔷 Key Idea

• Every element of A is already present in U (because U contains all elements)
• So the elements common to U and A are exactly the elements of A

---

🔷 Simple Example

Let:

• U = {1, 2, 3, 4, 5}
• A = {2, 4}

Now:

U \cap A = \{2, 4\} = A

---

🔷 Final Result

\boxed{U \cap A = A}

---

✅ Correct Answer

(ii) A

---

✨ Exam Tip (Important Law)

Intersection of any set with the universal set gives the set itself.

This explanation is perfect for MCQs and short-answer questions.

If you want:

• More law-based questions
• De Morgan’s laws explanation
• One-page theory revision

Just tell me 😊

Here are clear theory + simple examples for each operation on sets (Perfect for Plus One exams) ✅

---

🔷 Operations on Sets – Theory & Examples

Let
A = {1, 2, 3}
B = {3, 4, 5}
U = {1, 2, 3, 4, 5, 6}

---

1️⃣ Union (A ∪ B)

📘 Theory

The union of two sets A and B is the set of all elements which are in A or in B or in both.

Symbol:

A \cup B = \{x : x \in A \text{ or } x \in B\}

✏️ Example

A = {1, 2, 3}
B = {3, 4, 5}

A \cup B = \{1, 2, 3, 4, 5\}

👉 Note: Common elements are written only once.

---

2️⃣ Intersection (A ∩ B)

📘 Theory

The intersection of two sets is the set of common elements present in both A and B.

Symbol:

A \cap B = \{x : x \in A \text{ and } x \in B\}

✏️ Example

A = {1, 2, 3}
B = {3, 4, 5}

A \cap B = \{3\}

👉 Only common element is written.

---

3️⃣ Difference (A − B)

📘 Theory

The difference of two sets A and B is the set of elements which are in A but not in B.

Symbol:

A - B = \{x : x \in A \text{ and } x \notin B\}

✏️ Example

A = {1, 2, 3}
B = {3, 4, 5}

A - B = \{1, 2\}

👉 3 is removed because it is common with B.

---

4️⃣ Complement (A′)

📘 Theory

The complement of a set A is the set of elements in the universal set U but not in A.

Symbol:

A' = U - A

✏️ Example

U = {1, 2, 3, 4, 5, 6}
A = {1, 2, 3}

A' = \{4, 5, 6\}

---

🌟 Easy Memory Trick (Exam)

• Union (∪) → OR
• Intersection (∩) → AND
• Difference (−) → IN FIRST, NOT IN SECOND
• Complement (′) → IN U, NOT IN A

---

If you want next:

• ✔️ Venn diagram explanation
• ✔️ Exam MCQs
• ✔️ Verification problems
• ✔️ One-page formula sheet

Just tell me 😊

Sure 😊
I’ll explain very simply, step-by-step, with easy words + small examples (exam-friendly).

---

🔷 Operations on Sets (Very Simple)

Think of a set as a box of numbers.

Let
A = {1, 2, 3}
B = {3, 4}
U = {1, 2, 3, 4, 5}

---

1️⃣ Union (A ∪ B)

🧠 Meaning (Very Simple)

Union means “ALL together”
Take everything from A and B.
Don’t repeat common elements.

✏️ Example

A = {1, 2, 3}
B = {3, 4}

A \cup B = \{1, 2, 3, 4\}

👉 3 is common, write it only once.

📌 Key word: OR

---

2️⃣ Intersection (A ∩ B)

🧠 Meaning

Intersection means “COMMON”
Take only elements present in both sets.

✏️ Example

A = {1, 2, 3}
B = {3, 4}

A \cap B = \{3\}

👉 Only 3 is common.

📌 Key word: AND

---

3️⃣ Difference (A − B)

🧠 Meaning

Difference means “REMOVE”
Take elements in A and remove those also in B.

✏️ Example

A = {1, 2, 3}
B = {3, 4}

A - B = \{1, 2\}

👉 3 is removed because it is in B.

📌 Trick: First set only!

---

4️⃣ Complement (A′)

🧠 Meaning

Complement means “OUTSIDE A”
Take elements from Universal set U which are not in A.

✏️ Example

U = {1, 2, 3, 4, 5}
A = {1, 2, 3}

A' = \{4, 5\}

👉 Elements inside A are removed from U.

📌 Formula:

A' = U - A

---

🌟 Easy Daily-Life Example

👧 Girls in class = A
👦 Boys in class = B
👨‍👩‍👧‍👦 All students = U

• A ∪ B → All students
• A ∩ B → No one (empty set)
• A − B → Only girls
• A′ → Students who are not girls (boys)

---

📝 Exam Memory Box

Operation	Meaning
∪	All
∩	Common
−	Remove
′	Outside

---