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
Right, let's go. a) Step 1: List the elements of the universal set U. U = \1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12\ Step 2: Determine the elements of set A. A = \x : x is divisible by 3\ The numbers in U divisible by 3 are 3, 6, 9, 12. A = \3, 6, 9, 12\ Step 3: Determine the elements of set B. B = \x : x is a factor of 18\ The factors of 18 are 1, 2, 3, 6, 9, 18. From U, these are 1, 2, 3, 6, 9. B = \1, 2, 3, 6, 9\ Step 4: Determine the elements of set C. C = \x : x is even and not divisible by 5\ Even numbers in U: \2, 4, 6, 8, 10, 12\. Numbers in U divisible by 5: \5, 10\. Numbers that are even and not divisible by 5: \2, 4, 6, 8, 12\. C = \2, 4, 6, 8, 12\ Step 5: Find the union of sets B and C, B C. B C = \1, 2, 3, 6, 9\ \2, 4, 6, 8, 12\ B C = \1, 2, 3, 4, 6, 8, 9, 12\ Step 6: Find the intersection of set A and (B C), A (B C). A (B C) = \3, 6, 9, 12\ \1, 2, 3, 4, 6, 8, 9, 12\ The common elements are 3, 6, 9, 12. A (B C) = \3, 6, 9, 12\ Step 7: Find the number of elements in A (B C), denoted as n[A (B C)]. The set A (B C) has 4 elements. n[A (B C)] = 4 The value of n[A (B C)] is 4. b) Step 1: Rewrite the equation using exponent rules. The given equation is 3^2y - 4 × 3^y + 3 = 0. We can write 3^2y as (3^y)^2. (3^y)^2 - 4 × 3^y + 3 = 0 Step 2: Use a substitution to transform the equation into a quadratic form. Let u = 3^y. Substitute u into the equation: u^2 - 4u + 3 = 0 Step 3: Solve the quadratic equation for u. Factor the quadratic equation: (u - 1)(u - 3) = 0 This gives two possible values for u: u - 1 = 0 u = 1 u - 3 = 0 u = 3 Step 4: Substitute back u = 3^y and solve for y. Case 1: u = 1 3^y = 1 Since 3^0 = 1, we have: y = 0 Case 2: u = 3 3^y = 3 Since 3^1 = 3, we have: y = 1 The solutions for y are 0 and 1.