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 based on the provided Venn diagram. The elements in the Venn diagram are: Universal Set U = \a, b, c, d, e, f, g, i, m\ Set G = \a, b, c, g, i\ Set H = \c, d, e, g\ Set K = \f, g, i\ a) What is the cardinal number of the universal set U? Step 1: List all distinct elements in the universal set U from the Venn diagram. U = \a, b, c, d, e, f, g, i, m\ Step 2: Count the number of elements in U. |U| = 9 The cardinal number of the universal set U is 9. b) Write the set (G K) by listing method. Step 1: Identify the elements in set G. G = \a, b, c, g, i\ Step 2: Identify the elements in set K. K = \f, g, i\ Step 3: Find the intersection of G and K, which are the elements common to both sets. G K = \g, i\ The set (G K) is \g, i\. c) Prove that: G H = G H Step 1: Find G H. G H = \a, b, c, g, i\ \c, d, e, g\ G H = \a, b, c, d, e, g, i\ Step 2: Find the complement of G H, denoted as G H. G H = U - (G H) G H = \a, b, c, d, e, f, g, i, m\ - \a, b, c, d, e, g, i\ G H = \f, m\ Step 3: Find the complement of G, denoted as G. G = U - G G = \a, b, c, d, e, f, g, i, m\ - \a, b, c, g, i\ G = \d, e, f, m\ Step 4: Find the complement of H, denoted as H. H = U - H H = \a, b, c, d, e, f, g, i, m\ - \c, d, e, g\ H = \a, b, f, i, m\ Step 5: Find the intersection of G and H, denoted as G H. G H = \d, e, f, m\ \a, b, f, i, m\ G H = \f, m\ Step 6: Compare the results from Step 2 and Step 5. Since G H = \f, m\ and G H = \f, m\, Therefore, G H = G H is proven. d) What percent of the elements of universal set U have (G H K)? Step 1: Find the union of sets G, H, and K. G H K = \a, b, c, g, i\ \c, d, e, g\ \f, g, i\ G H K = \a, b, c, d, e, f, g, i\ Step 2: Find the cardinal number of G H K. |G H K| = 8 Step 3: Recall the cardinal number of the universal set U. |U| = 9 Step 4: Calculate the percentage of elements in U that are in (G H K). Percentage = (|G H K|)/(|U|) × 100\% Percentage = (8)/(9) × 100\% Percentage ≈ 0.8889 × 100\% Percentage ≈ 88.89\% The percentage of elements of universal set U that have (G H K) is approximately 88.89\%.