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
Eharuner, let's knock this out. The question asks for the condition under which A B = B. Step 1: Understand the meaning of A B = B. The intersection of two sets, A B, contains all elements that are common to both set A and set B. If A B = B, it means that all elements of set B are also present in set A. If every element of B is also an element of A, then B is a subset of A. Step 2: Express "B is a subset of A" using set notation. This is written as B A. The notation B A typically means B is a proper subset of A (meaning B is a subset of A, and B is not equal to A). However, in multiple-choice questions, B A is often used to represent B A when B A is not an option. Step 3: Evaluate the given options. A. A = B^c: This means A is the complement of B. If A is the complement of B, then A and B have no common elements, so A B = . This is only equal to B if B is the empty set, which is not generally true. B. A B: This means A is a subset of B. If A is a subset of B, then A B = A. For this to be equal to B, we would need A = B. This is not the general condition for A B = B. C. A^c = B: This is the same as option A, just written differently. It implies A B = , which is not generally true. D. B A: This means B is a subset of A (or a proper subset). If B is a subset of A, then all elements of B are also in A. Therefore, the elements common to both A and B are precisely the elements of B. So, A B = B. This condition is equivalent to B A. The condition A B = B is true if and only if B is a subset of A. The final answer is D. B A. What's next? 📸