Question: Fifty-three of the fifty-five BSIT students of CSTC – Main Campus are taking at least one of the mathematics subjects, Algebra, Trigonometry, and Calculus. Suppose that: 24 taking Algebra, 26 taking Trigonometry, 20 taking Calculus, 5 taking Algebra and Trigonometry, 7 taking Algebra and Calculus, and 8 taking Trigonometry and Calculus.
a. Find the number of students who are taking all three mathematics subjects.
b. How many students are taking Algebra as their only mathematics subject?
c. How many students are taking Calculus and Trigonometry but not Algebra? Create a Venn diagram to represent this (identify the shaded part).
d. Determine the number of students not taking the three mathematics subjects.
1. **State the problem:**
We have 55 students, 53 of whom take at least one of Algebra (A), Trigonometry (T), or Calculus (C).
Given:
- $|A|=24$
- $|T|=26$
- $|C|=20$
- $|A \cap T|=5$
- $|A \cap C|=7$
- $|T \cap C|=8$
- $|A \cup T \cup C|=53$
We need to find:
(a) $|A \cap T \cap C|$
(b) Number taking only Algebra
(c) Number taking Calculus and Trigonometry but not Algebra
(d) Number not taking any of the three subjects
2. **Formula used:**
The Inclusion-Exclusion Principle for three sets:
$$|A \cup T \cup C| = |A| + |T| + |C| - |A \cap T| - |A \cap C| - |T \cap C| + |A \cap T \cap C|$$
3. **Find $|A \cap T \cap C|$:**
Substitute known values:
$$53 = 24 + 26 + 20 - 5 - 7 - 8 + |A \cap T \cap C|$$
Simplify:
$$53 = 70 - 20 + |A \cap T \cap C|$$
$$53 = 50 + |A \cap T \cap C|$$
Isolate $|A \cap T \cap C|$:
$$|A \cap T \cap C| = 53 - 50 = 3$$
4. **Find number taking only Algebra:**
Only Algebra means those in $A$ but not in $T$ or $C$.
Use:
$$|A| = |A \text{ only}| + |A \cap T| + |A \cap C| - |A \cap T \cap C|$$
Rearranged:
$$|A \text{ only}| = |A| - |A \cap T| - |A \cap C| + |A \cap T \cap C|$$
Substitute values:
$$|A \text{ only}| = 24 - 5 - 7 + 3 = 15$$
5. **Find number taking Calculus and Trigonometry but not Algebra:**
This is:
$$|T \cap C| - |A \cap T \cap C| = 8 - 3 = 5$$
6. **Find number not taking any of the three subjects:**
Total students = 55
Taking at least one subject = 53
So:
$$\text{Not taking any} = 55 - 53 = 2$$
---
**Final answers:**
- (a) $|A \cap T \cap C| = 3$
- (b) Algebra only = 15
- (c) Calculus and Trigonometry but not Algebra = 5
- (d) Not taking any = 2
---
**Venn diagram explanation for (c):**
The shaded region is the intersection of Calculus and Trigonometry excluding Algebra, which corresponds to the 5 students found in step 5.