সংহতি | Sets | Lesson 2 | Exercise 2.3 | Class 9 Advanced Mathematics Solution Assamese Medium | Class 9 Adv Math's Solution Assam | নৱম শ্ৰেণীৰ উচ্চ গণিতৰ প্ৰশ্ন উত্তৰ |

পাঠ ২

সংহতি

Sets

Exercise 2.3

1. ধৰা, A = {1, 3, 5, 6}, B = {3, 4, 6} আৰু C = {3, 7, 9} আৰু U = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10}

তলৰ বিজীয় প্ৰক্ৰিয়াবোৰ উলিওৱাঃ

(a) (A∪B)∩C

উত্তৰঃ

A∪B = {1, 3, 4, 5, 6}

আৰু C = {3, 7, 9}

∴ (A∪B)∩C = {3}

(b) (C - A)′∪ B

উত্তৰঃ

(C - A) = {7, 9} 

∴ (C - A)՛ = {1, 2, 3, 4, 5, 6, 8, 10}

∴ (C - A)′∪ B = {1, 2, 3, 4, 5, 6, 8, 10}

(c) (A - B) ∪ (B - A)

উত্তৰঃ A - B = {{1, 5}

B - A = {4, 6}

∴ (A - B) ∪ (B - A) = {1, 4, 5, 6}

(d) (A ∪ B) - C

উত্তৰঃ A ∪ B = {1, 3, 4, 5, 6}

∴ (A ∪ B) - C = {1, 4, 5, 6}

2. দিয়া আছে; U = {1, 2, 3,......10}

A = {x: x এটা অখণ্ড সংখ্যা, 1⩽ x < 6}

B = {x: x এটা যুগ্ম সংখ্যা, 4 ⩽ x < 9}

C = {x: x এটা অখণ্ড সংখ্যা, 1 ⩽ x² ⩽ 16

(i) (A ∪ B) ∩ C    (ii) (A - C) ∩ B    (iii) (A - B)' ∪ C    (iv) (A ∩ B) ∪ C'

উত্তৰঃ 

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

B = {4, 6, 8}

C = {1, 2, 3, 4}

∴ U = {1, 2, 3, 4, 5, 6, 8}

(i) A ∪ B ∩ C

A ∪ B = {1, 2, 3, 4, 5, 6, 8}

∴  (A ∪ B) ∩ C = {1, 2, 3, 4}

(ii) (A - C) ∩ B

A - C = {5}

∴ (A - C) ∩ B = ϕ

(iii) (A - B)' ∪ C

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

(A - B)' = {4, 6, 8}

∴ (A - B)' ∪ C = {1, 2, 3, 4, 6, 8}

(iv) (A ∩ B) ∪ C'

A ∩ B = {4}

C' = {5, 6, 8}

∴ (A ∩ B) ∪ C' = {4, 5, 6, 8}

3. যদি A_i = {i + 1, ( i + 1)², ( i + 1)³}  য'ত i = 1, 2, 3, 4, 5 তেন্তে A₁, A₂, A₃ সংহতিবোৰ লিখা পৰীক্ষা কৰা যে ইহঁত যুৰীয়াকৈ অসংযুক্ত।

উত্তৰঃ 

A₁ 

= {1 + 1, (1 + 1)², ( 1 + 1)³}

= {2, 2², 2³}

= (2, 4, 8)

A₂ 

= {2 + 1, (2 + 1)², (2 + 1)³}

= (3, 3² , 3³)

= (3, 9, 27)

A₃ 

= {3 + 1, (3 + 1)², (3 + 1)³}

= (4, 4², 4³)

= (4, 16, 64)

A₁∩A₂ = ϕ

A₂∩A₃ = ϕ

আৰু A₁∩A₃ = ϕ

∴ প্ৰমাণত হ'ল যে ইহঁত যুৰীয়াকৈ অসংযুক্ত।

4. A = {x| x ∈ N, 1 ⩽ x < 6}

B = {x| x ∈ Z, 3 ⩽ x < 8}

C = {1, 3, 5}

আৰু U = {x| x ∈ Z, 0 ⩽ x < 10} দিয়া আছে।

উত্তৰঃ 

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

B = {3, 4, 5, 6, 7}

C = {1, 3, 5}

আৰু U = {0, 1, 2, 3, 4, 5, 6, 7, 8, 9}

∴ A' = {0, 6, 7, 8, 9}

B' = {0, 1, 2, 8, 9}

আৰু C' = {0, 2, 4, 6, 7, 8, 9}

(i) (A∪B)' = A'∩B'

L.H.S; 

A∪B = {1, 2, 3, 4, 5, 6, 7}

(A∪B)' = {0, 8, 9}

R.H.S; 

A'∩B' = {0, 8, 9}

∴ L.H.S = R.H.S

(ii) (A∩B)' = A'∪B'

L.H.S; 

A∩B = {3, 4, 5}

(A∩B)' = {0, 1, 2, 6, 7, 8, 9}

R.H.S;

A'∪B' = {0, 1, 2, 6,7, 8, 9}

∴ L.H.S = R.H.S

(iii) A∪A' = U

L.H.S; 

A∪A' = {0, 1, 2, 3, 4, 5, 6, 7, 8, 9}

R.H.S 

U = {0, 1, 2, 3, 4, 5, 6, 7, 8, 9}

∴ L.H.S = R.H.S

(iv) A∩A' = ϕ

উত্তৰঃ 

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

আৰু A' = {0, 6, 7, 8, 9}

∴ A∩A' = ϕ

(v) A∩(B∪C) = (A∩B)∪(A∩C)

উত্তৰঃ 

(B∪C) = {1, 3, 4, 5, 6, 7}

A∩(B∪C) = {1, 3, 4, 5}

আকৌ,

A∩B = {3, 4, 5}

A∩C = {1, 3, 5}

(A∩)∪(A∩C) = {1, 3, 4, 5}

∴ L.H.S = R.H.S

5. N স্বাভাৱিক সংখ্যাৰ সংহতি; আৰু a ∈ N। যদি aN = {ax| x ∈ N, x ⩽ 10} তেন্তে 

উত্তৰঃ 

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

2N = {2, 4, 6, 8, 10, ........}

3N = {3, 6, 9, 12, 15, .........}

5N = {5, 10, 15, 20, .........}

∴

(i) N∩2N আৰু 3N∪5N 

N∩2N = {2, 4, 6, 8,.......}

3N∪5N = {3, 5, 6, 10, 9, 15,.......}

(ii) দেখুওৱা যে 2N∪3N ≠ 5N

L.H.S

2N∪3N = {2, 3, 4, 6, 8, 9,......}

আৰু 

5N = {5,10, 15, ........}

∴ L.H.S ≠ R.H.S

(iii) কোনটো সত্য?

(a) 2N ⊂ N    (b) N ⊂ 2N    (c) 2N = N

উত্তৰঃ (a) 2N ⊂ N

- Hiru Moni Bora

Post Id: DABP003404