Question

A survey was conducted at a coaching institution and itswas found that there were 34 students who appeared in MAT. There were 37 students who appeared in CAT of which 17 students appeared in MAT. 30 students appeared in XAT of which 13 students appeared in MAT. Of the XAT applicants (i.e., appeared students) 14 appeared in CAT and these 6 appeared in MA'T. How many students appeared in CAT but not in MAT or XAT? (a) 9 (b) 10 (c) 12 (d) none of these

Short answer

Answer: 12 students appeared in CAT but not in MAT or XAT.

Step-by-step solution

Define the sets and the given data

Let's define the following sets: - Set A: Students who appeared in MAT - Set B: Students who appeared in CAT - Set C: Students who appeared in XAT Given data: - |A| = 34 - |B| = 37 - |A ∩ B| = 17 - |C| = 30 - |A ∩ C| = 13 - |B ∩ C| = 14 - |A ∩ B ∩ C| = 6

Applying the Principle of Inclusion-Exclusion

Using the principle of inclusion-exclusion, we have the formula for the total number of students (|U|) who appeared in at least one of the three exams: |U| = |A| + |B| + |C| - |A ∩ B| - |A ∩ C| - |B ∩ C| + |A ∩ B ∩ C| Substitute the given values: |U| = 34 + 37 + 30 - 17 - 13 - 14 + 6 = 63

Calculate the number of students who appeared in CAT but not in MAT or XAT

We are looking for the number of students who appeared in CAT (set B) but not in MAT (set A) or XAT (set C), which is given by: |B - (A ∪ C)| To find this value, we can calculate |B ∩ (A ∪ C)| and then subtract it from |B|: - |B ∩ (A ∪ C)| = |B ∩ A| + |B ∩ C| - |A ∩ B ∩ C| = 17 + 14 - 6 = 25 Now, subtract this value from |B|: |B - (A ∪ C)| = |B| - |B ∩ (A ∪ C)| = 37 - 25 = 12 So, there are 12 students who appeared in CAT but not in MAT or XAT. The correct answer is (c) 12.

Key concepts

Principle of Inclusion-Exclusion

The Principle of Inclusion-Exclusion is a fundamental concept in probability and set theory. It is used to calculate the size of the union of multiple sets. This principle helps us avoid double-counting elements that appear in more than one set.

In mathematical terms, if you have sets A, B, and C, the principle is expressed as follows:

• |A ∪ B ∪ C| = |A| + |B| + |C| - |A ∩ B| - |A ∩ C| - |B ∩ C| + |A ∩ B ∩ C|.

Each of these terms represent different overlaps:

• |A|, |B|, and |C| are the sizes of each set individually.
• |A ∩ B|, |A ∩ C|, and |B ∩ C| represent the intersections of two sets, where elements are counted twice if you only sum up individual sets.
• |A ∩ B ∩ C| is the intersection of all three sets, counted once because they've been subtracted three times in the intersection calculations.

This principle ensures that each element is only counted once across the union of the sets. It is extremely useful when dealing with surveys or problems involving overlapping groups, as it provides a structured way to calculate how many items belong uniquely to any number of sets in a simple and efficient manner. In our problem, it allowed us to calculate the total number of students attending the tests accurately without double counting.

Set Theory

Set theory is the branch of mathematical logic that studies sets, which are collections of objects. In a Venn diagram, each set is usually represented as a circle or an ellipse, with overlaps representing intersections and unions.

In our problem, we dealt with three sets:

• Set A: Students who appeared in MAT
• Set B: Students who appeared in CAT
• Set C: Students who appeared in XAT

The goal was to find how many students belonged uniquely to certain combinations of these sets. We use basic set operations such as:

• Intersection (∩): Represents elements common to both sets, as with students appearing in both CAT and MAT.
• Union (∪): Refers to elements that belong to at least one of the sets, like students appearing in any of the exams.

Set theory allows us to handle these combinations systematically, helping to answer questions about shared and unique elements within any group. By using set theory, complex relationships between different groups or categories can be visualized and solved, as demonstrated by the problem solution.

Problem Solving Strategies

Great problem-solving involves breaking down the problem into understandable chunks, using logical reasoning, and systematically applying mathematical principles.

Here’s a strategic approach to problem-solving using our exercise:

• **Define the Sets:** Clearly define the universe of discourse with the relevant sets and provided details.
• **Understand the Data:** Translate the word problem into mathematical language using numbers and set operations like intersections and unions.
• **Apply Mathematical Tools:** Utilize known principles and formulas, such as the Principle of Inclusion-Exclusion, to solve for unknowns.
• **Verify Your Answer:** Reevaluate the solution to ensure consistency and validity, ensuring you've correctly interpreted the overlaps and unique elements.

By following such steps, complex problems become manageable, errors are minimized, and solutions become accessible in a coherent manner. This methodical approach ensures all possibilities are considered, and the most accurate solution is reached efficiently. Effective use of these strategies not only solves the current problem but also builds a framework for tackling future problems.