This mathematics problem involves applying core mathematical principles and formulas. Below you will find a complete step-by-step solution with detailed explanations for each step, helping you understand not just the answer but the method behind it.
ScanSolve AI Answer
Step 1: Solve Question 7. The question states: "যদি A B = B হয়, তবে—" (If A B = B, then—). If the intersection of set A and set B is equal to set B, it means that all elements of B are also elements of A. This is the definition of B being a subset of A. Therefore, B A. The correct option is B. B Step 2: Solve Question 8. The question states: "A ও B দুটি বিচ্ছেদ (disjoint) সেট হলে, n(A B) = " (If A and B are two disjoint sets, then n(A B) = ). For any two finite sets A and B, the number of elements in their union is given by the formula: n(A B) = n(A) + n(B) - n(A B) If A and B are disjoint sets, it means they have no common elements. Therefore, their intersection is the empty set, A B = , and the number of elements in their intersection is n(A B) = 0. Substituting n(A B) = 0 into the formula: n(A B) = n(A) + n(B) - 0 n(A B) = n(A) + n(B) The correct option is A. A Step 3: Solve Question 9. The question states: "যে-কোনো দুটি সেট A ও B-এর ক্ষেত্রে, n(A) + n(B) - n(A B) = " (For any two sets A and B, n(A) + n(B) - n(A B) = ). This is the fundamental formula for the number of elements in the union of two finite sets. n(A) + n(B) - n(A B) = n(A B) The correct option is A. A Step 4: Solve Question 10. The question states: "A U = U-এর দ্বৈত অভেদ হবে—" (The dual identity of A U = U will be—). To find the dual of a set identity, we interchange the union operator () with the intersection operator (), and the universal set (U) with the empty set (). Given identity: A U = U 1. Replace with : A U 2. Replace U with : A 3. Replace U with on the right side: So, the dual identity is A = . The correct option is D. D Step 5: Solve Question 11. The question states: "A (B C) = (A B) (A C) অভেদের দ্বৈত অভেদ হয়—" (The dual identity of A (B C) = (A B) (A C) is—). To find the dual of a set identity, we interchange the union operator () with the intersection operator (), and vice versa. Given identity: A (B C) = (A B) (A C) 1. Replace with : A () and () () 2. Replace with : ( B C ) and ( A B ) and ( A C ) Applying these changes to the given identity: Left side: A (B C) becomes A (B C) Right side: (A B) (A C) becomes (A B) (A C) So, the dual identity is A (B C) = (A B) (A C). This is the distributive law of intersection over union. The correct option is A. A