Question

Which of the following collections is not a set?

• A. The collection of natural numbers between 2 and 20
• B. The collection of numbers which satisfy the equation \(x^2-5x+6\)
• C. The collection of prime numbers between 1 and 100
• D. The collection of all brilliant students in a class

Answer 1

The Correct Option is D

We need to determine which of the given collections is not a set. 

Definition of a Set: A set is a well-defined collection of distinct objects. A collection is well-defined if there is no ambiguity about whether an element belongs to it or not.

Step 1: Analyze Each Option

Option 1: The collection of natural numbers between 2 and 20.

Natural numbers between 2 and 20 are: \( \{3, 4, 5, ..., 19\} \). Since the elements are clearly defined, this is a set.

Option 2: The collection of numbers which satisfy the equation \(x^2 - 5x + 6 = 0\).

Solving the equation:

\[ x^2 - 5x + 6 = 0 \]

\[ (x - 2)(x - 3) = 0 \]

\[ x = 2, 3 \]

The collection is \(\{2, 3\}\), which is well-defined. Thus, this is a set.

Option 3: The collection of prime numbers between 1 and 100.

Prime numbers between 1 and 100 are clearly defined (e.g., \(2, 3, 5, 7, ...\)), so this is a set.

Option 4: The collection of all brilliant students in a class.

The term "brilliant students" is subjective and can vary based on different opinions or criteria. Since there is no clear definition, this collection is not a set.

Final Answer: The collection of all brilliant students in a class.

Answer 2

The concept of a set in mathematics refers to a collection of well-defined and distinct objects. For a collection to be considered a set, it must be possible to clearly determine whether or not an element belongs to the collection.

Let's analyze each option to identify which is not a set:

• The collection of natural numbers between 2 and 20: This is a well-defined set, as it's clear which numbers are included (3, 4, 5, ..., 19).
• The collection of numbers which satisfy the equation \(x^2-5x+6\): Solving the equation, we find the roots are \(x=2\) and \(x=3\). Therefore, this collection is well-defined and hence a set.
• The collection of prime numbers between 1 and 100: Prime numbers in this range are well-defined (2, 3, 5, ..., 97), making this a set.
• The collection of all brilliant students in a class: The term "brilliant" is subjective and varies based on criteria like intelligence, academic performance, etc. This vagueness means it is not a well-defined collection and hence not a set.

Thus, the collection of all brilliant students in a class is not a set due to the lack of a clear definition.