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
Here are the solutions to the questions: 1. a) i) List the elements in set 'A' and 'B'. Step 1: Define the universal set U. U = \Odd numbers less than 19\ U = \1, 3, 5, 7, 9, 11, 13, 15, 17\ Step 2: Define set A. A = \Factors of 15\ The factors of 15 are the numbers that divide 15 exactly. A = \1, 3, 5, 15\ Step 3: Define set B. B = \Prime numbers\ (within U) Prime numbers in U are numbers greater than 1 that are only divisible by 1 and themselves. B = \3, 5, 7, 11, 13, 17\ The elements in set A are \1, 3, 5, 15\ and in set B are \3, 5, 7, 11, 13, 17\. ii) Find the set of A B and A B. Step 1: Find the intersection A B. A B contains elements common to both A and B. A = \1, 3, 5, 15\ B = \3, 5, 7, 11, 13, 17\ A B = \3, 5\ Step 2: Find the union A B. A B contains all elements in A or B (or both). A B = \1, 3, 5, 7, 11, 13, 15, 17\ The set A B is \3, 5\ and A B is \1, 3, 5, 7, 11, 13, 15, 17\. iii) Illustrate U, A and B on a Venn diagram. Step 1: Draw a rectangle to represent the universal set U. Step 2: Draw two overlapping circles inside the rectangle, one for set A and one for set B. Step 3: Place the elements in the correct regions: • A B = \3, 5\ (in the overlap) • Elements in A only: A B = \1, 15\ • Elements in B only: B A = \7, 11, 13, 17\ • Elements in U but not in A or B: U (A B) = \9\ ` U +---------------------------------+ | +-------+ +-------+ | | | A | | B | | | | 1,15| | 7,11,| | | | +--+---+--+ | | | | |3,5| | | | | | +--+---+--+ | | | | | |13,17| | | +-------+ +-------+ | | | | 9 | +---------------------------------+ ` iv) Shade the region for prime factors of 15 on the Venn diagram. Step 1: Identify the prime factors of 15. The prime factors of 15 are 3 and 5. Step 2: Locate these elements on the Venn diagram. The elements 3 and 5 are in the intersection of sets A and B (A B). Step 3: Shade the region representing A B. 1. b) Find the value of x if 3^x-1 = 27. Step 1: Express 27 as a power of 3. 27 = 3^3 Step 2: Rewrite the equation with the same base. 3^x-1 = 3^3 Step 3: Equate the exponents. x - 1 = 3 Step 4: Solve for x. x = 3 + 1 x = 4 The value of x is 4. 2. a) Determine the combination of GH¢20, GH¢50, GH¢100, GH¢200 and GH¢500 notes to make GH¢1,000,000. This question asks for a combination. Here is one possible combination: Step 1: Use GH¢500 notes. Number of GH¢500 notes = GH¢500,000GH¢500 = 1,000 notes. Remaining amount = GH¢1,000,000 - GH¢500,000 = GH¢500,000. Step 2: Use GH¢200 notes. Number of GH¢2