Question 6. a) Write the cardinalities of the given sets in set notational forms. Show the above information in Venn-diagram.

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, $n(M \setminus E) = 18$.
Students who got 'A+' grade in English only, $n(E \setminus M) = 25$.
Students who did not get 'A+' grade in these two subjects, $n(M \cup E)' = 7$.

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:

$$n(U) = n(M \setminus E) + n(E \setminus M) + n(M \cap E) + n(M \cup E)'$$

Substitute the given values:

$$54 = 18 + 25 + n(M \cap E) + 7$$ $$54 = 50 + n(M \cap E)$$ $$n(M \cap E) = 54 - 50$$ $$n(M \cap E) = 4$$

a) (i) Write the cardinalities of the given sets in set notational forms.

• Total students: $n(U) = 54$
• Students with 'A+' in Mathematics only: $n(M \setminus E) = 18$
• Students with 'A+' in English only: $n(E \setminus M) = 25$
• Students with 'A+' in neither subject: $n(M \cup E)' = 7$
• Students with 'A+' in both subjects: $n(M \cap E) = 4$

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 ($M \cap E$) contains 4 students.
• The region outside both circles ($M \cup E)'$) 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:

$$n(M) = n(M \setminus E) + n(M \cap E)$$ $$n(M) = 18 + 4$$ $$n(M) = 22$$

The number of students who got 'A+' grade in Mathematics is $\boxed{\text{22 students}}$.

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 ($n(E')$):

$$n(E') = n(M \setminus E) + n(M \cup E)'$$ $$n(E') = 18 + 7$$ $$n(E') = 25$$

Next, find the number of students who got 'A+' grade not in Mathematics ($n(M')$):

$$n(M') = n(E \setminus M) + n(M \cup E)'$$ $$n(M') = 25 + 7$$ $$n(M') = 32$$

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 $\boxed{\text{7 less students}}$ who got 'A+' grade not in English than not in Mathematics.