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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Step 1: Solve Question 6. Given the sets: A = \1, 2, 3, 4\ B = \3, 4, 5, 6\ C = \2, 4, 6, 8\ a) Compute A B The union of two sets includes all elements present in either set. A B = \1, 2, 3, 4\ \3, 4, 5, 6\ A B = \1, 2, 3, 4, 5, 6\ b) Compute A C The intersection of two sets includes only the elements common to both sets. A C = \1, 2, 3, 4\ \2, 4, 6, 8\ A C = \2, 4\ c) Compute (A B) C First, calculate A B. From part (a), A B = \1, 2, 3, 4, 5, 6\. Next, find the intersection of this result with C. (A B) C = \1, 2, 3, 4, 5, 6\ \2, 4, 6, 8\ (A B) C = \2, 4, 6\ The answers for Question 6 are: *a) \1, 2, 3, 4, 5, 6\ *b) \2, 4\ *c) \2, 4, 6\ Step 2: Solve Question 7. Given the sets: A = \1, 2, 3\ B = \3, 4, 5\ Find the symmetric difference between A and B, denoted as A B. The symmetric difference is defined as A B = (A - B) (B - A). First, calculate A - B: A - B = \1, 2, 3\ - \3, 4, 5\ = \1, 2\ Next, calculate B - A: B - A = \3, 4, 5\ - \1, 2, 3\ = \4, 5\ Finally, find the union of these two results: A B = \1, 2\ \4, 5\ A B = \1, 2, 4, 5\ The answer for Question 7 is: \1, 2, 4, 5\ Step 3: Solve Question 8. Given the sets: A = \2, 4, 6, 8\ B = \4, 6, 10\ a) Compute A - B The set difference A - B contains elements that are in A but not in B. A - B = \2, 4, 6, 8\ - \4, 6, 10\ A - B = \2, 8\ b) Compute B - A The set difference B - A contains elements that are in B but not in A. B - A = \4, 6, 10\ - \2, 4, 6, 8\ B - A = \10\ The answers for Question 8 are: *a) \2, 8\ *b) \10\ Step 4: Solve Question 9. Given the sets: A = \1, 2, 3\ B = \2, 3, 4\ C = \3, 4, 5\ Compute the number of elements in A B C, denoted as |A B C|. First, find the union of all three sets: A B = \1, 2, 3\ \2, 3, 4\ = \1, 2, 3, 4\ (A B) C = \1, 2, 3, 4\ \3, 4, 5\ A B C = \1, 2, 3, 4, 5\ Now, count the number of elements in the resulting set: |A B C| = 5 The answer for Question 9 is: 5 Step 5: Solve Question 10. Given: Number of students enrolled in mathematics (|M|) = 100 Number of students enrolled in science (|S|) = 120 Number of students enrolled in both subjects (|M S|) = 80 Find the number of students enrolled in at least one of the subjects, which is |M S|. Use the Principle of Inclusion-Exclusion for two sets: |M S| = |M| + |S| - |M S| Substitute the given values: |M S| = 100 + 120 - 80 |M S| = 220 - 80 |M S| = 140 The answer for Question 10 is: 140 Step 6: Solve Question 11. Given the sets: A = \1, 2\ B = \2, 3\ C = \3, 4\ a) Compute (A B) C First, calculate A B: A B = \1, 2\ \2, 3\ = \1, 2, 3\ Next, find the union of this result with C: (A B) C = \1, 2, 3\ \3, 4\ (A B) C = \1, 2, 3, 4\ b) Compute A (B C) First, calculate B C: B C = \2, 3\ \3, 4\ = \2, 3, 4\ Next, find the union of A with this result: A (B C) = \1, 2\ \2, 3, 4\ A (B C) = \1, 2, 3, 4\ The answers for Question 11 are: *a) \1, 2, 3, 4\ *b) \1, 2, 3, 4\