3. (9) in a school with 300 people there is a finite math and chemistry class. in the school there are 110 students that are taking only one of these two classes. there are 50 students that are taking the chemistry class and there are 15 students taking both of the classes. determine the following.
(a) the number of students who are taking finite math.
(b) the number of students who are taking neither of these courses.
(c) the number of students not taking finite math.
4. suppose that u is a universal set and a, b, and c are subsets of u such that,
i. n(a∩b∩c)=45
ii. n(a∩c)=65
iii. n(b∩a)=60
iv. n((a∪b∪c)) = 30
v. n(a)=95
vi. n(b)=105
vii. n(c)=95
viii. n(u)=184
find,
(a) n(a∩b)
(b) n((a∪b))
(c) n((b∪c))

Question

3. (9) in a school with 300 people there is a finite math and chemistry class. in the school there are 110 students that are taking only one of these two classes. there are 50 students that are taking the chemistry class and there are 15 students taking both of the classes. determine the following.
(a) the number of students who are taking finite math.
(b) the number of students who are taking neither of these courses.
(c) the number of students not taking finite math.
4. suppose that u is a universal set and a, b, and c are subsets of u such that,
 i. n(a∩b∩c)=45
 ii. n(a∩c)=65
 iii. n(b∩a)=60
 iv. n((a∪b∪c)) = 30
 v. n(a)=95
 vi. n(b)=105
 vii. n(c)=95
 viii. n(u)=184
find,
(a) n(a∩b)
(b) n((a∪b))
(c) n((b∪c))

Accepted Answer

### 3.
#### (a)
# Explanation:
## Step1: Find students taking only chemistry
The number of students taking chemistry is 50 and 15 are taking both. So the number of students taking only chemistry is $50 - 15=35$.
## Step2: Find students taking only finite - math
The number of students taking only one of the two classes is 110. Since 35 are taking only chemistry, the number of students taking only finite - math is $110 - 35 = 75$.
## Step3: Find students taking finite - math
The number of students taking finite - math is the sum of those taking only finite - math and those taking both. So it is $75+15 = 90$.
# Answer:
90

#### (b)
# Explanation:
## Step1: Find students taking either class
The number of students taking chemistry is 50 and those taking finite - math is 90. But 15 are counted twice. So the number of students taking either class is $50 + 90-15=125$.
## Step2: Find students taking neither class
The total number of people in the school is 300. So the number of students taking neither class is $300 - 125=175$.
# Answer:
175

#### (c)
# Explanation:
## Step1: Total number of students
The total number of students in the school is 300.
## Step2: Number of students taking finite - math
We found in part (a) that 90 students are taking finite - math.
## Step3: Number of students not taking finite - math
The number of students not taking finite - math is $300 - 90 = 210$.
# Answer:
210

### 4.
#### (a)
# Explanation:
We know that $n(A\cap B)=n((A\cap B\cap C)\cup(A\cap B\cap C'))$. Also, we know that $n(A\cap B)$ is given as 60.
# Answer:
60

#### (b)
# Explanation:
## Step1: Use the formula for $n(A\cup B)$
$n(A\cup B)=n(A)+n(B)-n(A\cap B)$. Substituting $n(A) = 95$, $n(B)=105$ and $n(A\cap B)=60$, we get $n(A\cup B)=95 + 105-60=140$.
## Step2: Use the formula for the complement
$n((A\cup B)')=n(U)-n(A\cup B)$. Given $n(U)=184$, so $n((A\cup B)')=184 - 140 = 44$.
# Answer:
44

#### (c)
# Explanation:
## Step1: Use the formula for $n(B\cup C)$
$n(B\cup C)=n(B)+n(C)-n(B\cap C)$. We don't directly know $n(B\cap C)$. But we know that $n((A\cup B\cup C)') = 30$ and $n(U)=184$, so $n(A\cup B\cup C)=184 - 30=154$.
We also know that $n(A\cup B\cup C)=n(A)+n(B)+n(C)-n(A\cap B)-n(A\cap C)-n(B\cap C)+n(A\cap B\cap C)$.
Substituting the known values: $154=95 + 105+95-60 - 65 - n(B\cap C)+45$.
Simplifying gives $154=215 - n(B\cap C)$. So $n(B\cap C)=61$.
Then $n(B\cup C)=105 + 95-61=139$.
## Step2: Use the formula for the complement
$n((B\cup C)')=n(U)-n(B\cup C)$. So $n((B\cup C)')=184 - 139 = 45$.
# Answer:
45

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QUESTION IMAGE

Question

Explanation:

3.

(a)

Step1: Find students taking only chemistry

Step2: Find students taking only finite - math

Step3: Find students taking finite - math

Step1: Find students taking either class

Step2: Find students taking neither class

Step1: Total number of students

Step2: Number of students taking finite - math

Step3: Number of students not taking finite - math

Answer:

(b)