1. In a class of 50 students, 28 take Mathematics, 20 take Computer Science, and 12 take both. Using the principle of inclusion-exclusion, find the number of students who take either Mathematics or Computer Science.
2. Let [tex]$A=\{1,3,5\}$[/tex] and [tex]$B=\{2,3,4\}$[/tex].
(a) Find [tex]$A \cup B, A \cap B, A-B$[/tex], and [tex]$B-A$[/tex].
(b) Find the complement of [tex]$A$[/tex] with respect to the universal set [tex]$U=\{1,2,3,4,5,6\}$[/tex].
3. A company offers two services: mobile banking and SMS alerts. Out of 100 clients, 65 subscribe to mobile banking, 50 to SMS alerts, and 30 to both services.
(a) How many clients subscribe to at least one of the two services?
(b) How many clients subscribe to neither service?
4. Let [tex]$S=\{a, b, c\}$[/tex].
(a) List all elements of the power set [tex]$P (S)$[/tex].
(b) What is the cardinality of [tex]$P (S)$[/tex]?

Question

Anonymous · Answer

We calculated the number of students taking either Mathematics or Computer Science to be 36 using the principle of inclusion-exclusion. Additionally, the power set of set S contains 8 subsets. The calculations further clarify the relationships between different sets and their elements. 
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GinnyAnswer · Answer

Use the principle of inclusion-exclusion: ∣ M c u pC ∣ = ∣ M ∣ + ∣ C ∣ − ∣ M c a pC ∣ . 
 Substitute the given values: ∣ M c u pC ∣ = 28 + 20 − 12 . 
 Calculate the result: ∣ M c u pC ∣ = 36 . 
 The number of students who take either Mathematics or Computer Science is 36 ​ . 
 
 Explanation 
 
 Analyze the problem We are given the number of students taking Mathematics, Computer Science, and both. We need to find the number of students taking either Mathematics or Computer Science using the principle of inclusion-exclusion. 
 
 State the principle of inclusion-exclusion Let M be the set of students taking Mathematics, and C be the set of students taking Computer Science. We are given ∣ M ∣ = 28 , ∣ C ∣ = 20 , and ∣ M c a pC ∣ = 12 . We want to find ∣ M c u pC ∣ .

The principle of inclusion-exclusion states that ∣ M c u pC ∣ = ∣ M ∣ + ∣ C ∣ − ∣ M c a pC ∣ 
 
 Calculate the number of students taking either subject Substituting the given values, we have ∣ M c u pC ∣ = 28 + 20 − 12 = 48 − 12 = 36 
 
 State the answer Therefore, the number of students who take either Mathematics or Computer Science is 36.

Examples 
 In a school, you might want to know how many students are participating in sports or music clubs to plan resources effectively. The principle of inclusion-exclusion helps you avoid double-counting students who are in both.

Answer (2)

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