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.
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Here's the solution for question 2. First, we define the universal set and the three subsets A, B, and C. The universal set U is given as \1, 2, 3, , 20\. U = \1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20\ a) Define the sets A, B, and C. Step 1: Define Set A (all square numbers within U). Square numbers are numbers that are the product of an integer multiplied by itself. A = \x U x = n^2 for some integer n\ The square numbers in U are: 1^2 = 1 2^2 = 4 3^2 = 9 4^2 = 16 5^2 = 25 (This is greater than 20, so it's not in U) A = \1, 4, 9, 16\ Step 2: Define Set B (factors of 20 within U). Factors of 20 are numbers that divide 20 evenly. B = \x U 20 ÷ x is an integer\ The factors of 20 are: 1, 2, 4, 5, 10, 20 B = \1, 2, 4, 5, 10, 20\ Step 3: Define Set C (multiples of 4 within U). Multiples of 4 are numbers that can be divided by 4 evenly. C = \x U x = 4n for some integer n\ The multiples of 4 in U are: 4 × 1 = 4 4 × 2 = 8 4 × 3 = 12 4 × 4 = 16 4 × 5 = 20 4 × 6 = 24 (This is greater than 20, so it's not in U) C = \4, 8, 12, 16, 20\ b) Draw a Venn diagram to show the relationship between sets A, B, C. To draw the Venn diagram, we first find the intersections of the sets. Step 4: Find the intersections of the sets. • A B: Elements common to A and B. A = \1, 4, 9, 16\ B = \1, 2, 4, 5, 10, 20\ A B = \1, 4\ • A C: Elements common to A and C. A = \1, 4, 9, 16\ C = \4, 8, 12, 16, 20\ A C = \4, 16\ • B C: Elements common to B and C. B = \1, 2, 4, 5, 10, 20\ C = \4, 8, 12, 16, 20\ B C = \4, 20\ • A B C: Elements common to A, B, and C. From A B = \1, 4\ and C = \4, 8, 12, 16, 20\, the common element is 4. A B C = \4\ Step 5: Determine the elements unique to each region of the Venn diagram. • Elements in A B C: \4\ • Elements in A B only (not in C): \1\ • Elements in A C only (not in B): \16\ • Elements in B C only (not in A): \20\ • Elements in A only: A (A B A C) = \1, 4, 9, 16\ \1, 4, 16\ = \9\ • Elements in B only: B (A B B C) = \1, 2, 4, 5, 10, 20\ \1, 4, 20\ = \2, 5, 10\ • Elements in C only: C (A C B C) = \4, 8, 12, 16, 20\ \4, 16, 20\ = \8, 12\ • Elements outside A, B, and C (in U but not in A B C): A B C = \1, 2, 4, 5, 8, 9, 10, 12, 16, 20\ U (A B C) = \3, 6, 7, 11, 13, 14, 15, 17, 18, 19\ A Venn diagram would consist of a rectangle representing the universal set U, and three overlapping circles representing sets A, B, and C. The elements would be placed in the corresponding regions as follows: Region A B C: \4\ Region (A B) C: \1\ Region (A C) B: \16\ Region (B C) A: \20\ Region A (B C): \9\ Region B (A C): \2, 5, 10\ Region C (A B): \8, 12\ Region U (A B C) (outside all circles): \3, 6, 7, 11, 13, 14, 15, 17, 18, 19\ That's 2 down. 3 left today — send the next one.