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 to question 6. a): Let U be the set of all students. Let M be the set of students who got 'A+' grade in Mathematics. Let E be the set of students who got 'A+' grade in English. Given information: Total students, n(U) = 54. Students who got 'A+' grade in Mathematics only, n(M E) = 18. Students who got 'A+' grade in English only, n(E M) = 25. Students who did not get 'A+' grade in these two subjects, n(M E)' = 7. Step 1: Calculate the number of students who got 'A+' grade in both subjects. The total number of students is the sum of students in each disjoint region: n(U) = n(M E) + n(E M) + n(M E) + n(M E)' Substitute the given values: 54 = 18 + 25 + n(M E) + 7 54 = 50 + n(M E) n(M E) = 54 - 50 n(M E) = 4 a) (i) Write the cardinalities of the given sets in set notational forms. • Total students: n(U) = 54 • Students with 'A+' in Mathematics only: n(M E) = 18 • Students with 'A+' in English only: n(E M) = 25 • Students with 'A+' in neither subject: n(M E)' = 7 • Students with 'A+' in both subjects: n(M E) = 4 a) (ii) Show the above information in Venn-diagram. A Venn diagram would show two overlapping circles, M and E, inside a rectangle U. • The region for M only (left crescent) contains 18 students. • The region for E only (right crescent) contains 25 students. • The intersection region (M E) contains 4 students. • The region outside both circles (M E)') contains 7 students. a) (iii) How many students got 'A+' grade in Mathematics? The number of students who got 'A+' grade in Mathematics is the sum of those who got it in Mathematics only and those who got it in both subjects: n(M) = n(M E) + n(M E) n(M) = 18 + 4 n(M) = 22 The number of students who got 'A+' grade in Mathematics is 22 students. a) (iv) How many more or less students got 'A+' grade not in English than not in Mathematics? First, find the number of students who got 'A+' grade not in English (n(E')): n(E') = n(M E) + n(M E)' n(E') = 18 + 7 n(E') = 25 Next, find the number of students who got 'A+' grade not in Mathematics (n(M')): n(M') = n(E M) + n(M E)' n(M') = 25 + 7 n(M') = 32 Now, compare these two numbers. We want to know how many more or less students got 'A+' grade not in English than not in Mathematics. This is n(E') - n(M'): n(E') - n(M') = 25 - 32 n(E') - n(M') = -7 Since the result is negative, there are 7 less students who got 'A+' grade not in English than not in Mathematics. There are 7 less students who got 'A+' grade not in English than not in Mathematics. Send me the next one 📸