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
Hey Anup — ready when you are. Here are the solutions for question 1: First, let's define the sets based on the given information: Universal set U = \whole numbers from 1 to 10\ U = \1, 2, 3, 4, 5, 6, 7, 8, 9, 10\ Set A = \multiples of 3 within U\ A = \3, 6, 9\ Set B = \factors of 6 within U\ B = \1, 2, 3, 6\ Set C = \even numbers within U\ C = \2, 4, 6, 8, 10\ a) What is the cardinality for U? Step 1: Count the number of elements in set U. |U| = 10 The cardinality for U is 10. b) Illustrate the given information in a Venn-diagram. To illustrate in a Venn diagram, we first find the intersections: A B = \3, 6\ A C = \6\ B C = \2, 6\ A B C = \6\ Elements in regions: • Only A: \9\ • Only B: \1\ • Only C: \4, 8, 10\ • A B only (not C): \3\ • A C only (not B): • B C only (not A): \2\ • A B C: \6\ • Elements in U but outside A B C: \5, 7\ A Venn diagram would show a rectangle representing U, with three overlapping circles for A, B, and C. The elements would be placed in their respective regions as listed above. c) Is A-(A C) = (A B)-(B C), solve it. Step 1: Calculate the Left Hand Side (LHS): A-(A C). First, find A C: A C = \3, 6, 9\ \2, 4, 6, 8, 10\ = \2, 3, 4, 6, 8, 9, 10\ Now, find A-(A C): A-(A C) = \3, 6, 9\ - \2, 3, 4, 6, 8, 9, 10\ = Step 2: Calculate the Right Hand Side (RHS): (A B)-(B C). First, find A B: A B = \3, 6\ Next, find B C: B C = \1, 2, 3, 6\ \2, 4, 6, 8, 10\ = \1, 2, 3, 4, 6, 8, 10\ Now, find (A B)-(B C): (A B)-(B C) = \3, 6\ - \1, 2, 3, 4, 6, 8, 10\ = Step 3: Compare LHS and RHS. Since LHS = and RHS = , the equality holds. Therefore, A-(A C) = (A B)-(B C) is True. d) What are the groups A B C and B C? Write down with reason. Step 1: Find A B C. A B C = \3, 6, 9\ \1, 2, 3, 6\ \2, 4, 6, 8, 10\ A B C = \1, 2, 3, 4, 6, 8, 9, 10\ Reason: This set includes all elements from the universal set U that are either a multiple of 3, a factor of 6, or an even number (or any combination of these). Step 2: Find B C. B C = \1, 2, 3, 6\ \2, 4, 6, 8, 10\ B C = \1, 2, 3, 4, 6, 8, 10\ Reason: This set includes all elements from the universal set U that are either a factor of 6 or an even number (or both). Drop the next question!