Class 11 Set Theory Case Study: Student Club Memberships

Background: Students can choose to be members of these clubs, and some students are members of multiple clubs.

Information:

• There are 60 students in total at the school.
• 25 students are members of the Science Club (S).
• 20 students are members of the Mathematics Club (M).
• 15 students are members of the Arts Club (A).
• 10 students are members of both the Science and Mathematics Clubs (S ∩ M).
• 5 students are members of both the Science and Arts Clubs (S ∩ A).
• 8 students are members of both the Mathematics and Arts Clubs (M ∩ A).
• 3 students are members of all three clubs (S ∩ M ∩ A).

Question:

1. How many students are members of at least one of the three clubs (S ∪ M ∪ A)?
2. How many students are members of exactly two clubs?
3. How many students are not members of any club?
4. Can you identify the number of students who are only members of the Arts Club (A – (S ∪ M))?

Solution:

Given Information:

• Total number of students (universal set): n(U) = 60 students.
• Members of the Science Club (S): n(S) = 25 students.
• Members of the Mathematics Club (M): n(M) = 20 students.
• Members of the Arts Club (A): n(A) = 15 students.
• Members of both the Science and Mathematics Clubs (S ∩ M): n(S ∩ M) = 10 students.
• Members of both the Science and Arts Clubs (S ∩ A): n(S ∩ A) = 5 students.
• Members of both the Mathematics and Arts Clubs (M ∩ A): n(M ∩ A) = 8 students.
• Members of all three clubs (S ∩ M ∩ A): n(S ∩ M ∩ A)= 3 students.

Question 1: How many students are members of at least one of the three clubs (S ∪ M ∪ A)?

To find the number of students who are members of at least one of the clubs, we need to calculate the union of all three clubs:

$\mathrm{(S\; \cup \; M\; \cup \; A)\; =\; n(S)\; +n(M)\; +\; n(A)\; –\; n(S\cap M)\; –\; n(S\cap A)\; –\; n(M\cap A)\; +\; n(S\cap M\cap A) }$ $\mathrm{<span\; style="font-family:\; Georgia,\; \text{'}times\; new\; roman\text{'},\; \text{'}bitstream\; charter\text{'},\; Times,\; serif;">Substitute\; the\; given\; values:</span>}$

(S ∪ M ∪ A)

$=25+20+15-10-5-8+3$

n(S∪M∪A)=40

So, there are 40 students who are members of at least one of the three clubs.

Question 2: How many students are members of exactly two clubs?

To find the number of students who are members of exactly two clubs, we need to calculate the size of the union of sets representing students in two clubs but not in the third club:

Now, add these values:

Students in two clubs but not in the third club

=7

$+2+5=\mathrm{<span\; class="base"\; style="font-family:\; Georgia,\; \text{'}times\; new\; roman\text{'},\; \text{'}bitstream\; charter\text{'},\; Times,\; serif;"><span\; class="mord">14</span></span>}$

So, there are 14 students who are members of exactly two clubs.

Question 3: How many students are not members of any club?

To find the number of students who are not members of any club, we need to calculate the complement of the union of all three clubs in the universal set:

From the  Figure in question(2)

$n(U)-n(S\cup M\cup A)=60-40=20$

So, there are 20 students who are not members of any club.

Question 4: Can you identify the number of students who are only members of the Arts Club n(A) – n(S ∪ M)?

To find the number of students who are only members of the Arts Club, we need to calculate the set difference between the Arts Club and the union of the Science and Mathematics Clubs:

n(S∪M) = N(S) + n(M) – N(S∩M)

⇒ n(S∪M)  = 25 + 20 – 10 = 35

Now,

n(A) – n(S∪M) = 15 – 35 = -20

Since a negative number of students doesn’t make sense in this context, there are no students who are only members of the Arts Club n(A) – n(S ∪ M).

Please note that the answer in Question 4 is negative because there’s no scenario in the given information where students are only members of the Arts Club without being members of at least one of the other two clubs.

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