(a) What is Set? Explain use of Set with examples

 Set:

In mathematics and computer science, a set is a well-defined collection of distinct objects, considered as an object in its own right. These objects can be anything—numbers, symbols, or even other sets. The concept of a set provides a fundamental basis for various mathematical and computational operations.

Key Characteristics of a Set:

1. Distinct Elements: Every element in a set must be unique. There are no duplicate elements within a set.

2. Undefined Order: The elements in a set have no specified order. The notion of "first," "second," etc., is not applicable to sets.

3. Well-Defined Membership: An element either belongs to a set or does not. There is no ambiguity regarding the membership of elements.

Example:

Let's consider a set A that represents the prime numbers less than 10:

\[ A = \{2, 3, 5, 7\} \]

In this example:

- Each number in the set is unique.

- The order of the numbers does not matter.

- Each number is clearly defined as either belonging or not belonging to the set.

Use of Sets:

1. Mathematics:

   - Sets are fundamental in mathematical concepts like algebra, calculus, and statistics.

   - Operations like union, intersection, and complement are applied to sets to solve various mathematical problems.

2. Computer Science:

   - Sets are widely used in programming languages and data structures.

   - They provide an efficient way to manage and manipulate collections of data.

   - In Python, for example, the `set` data type is available to represent sets, and operations like union and intersection can be easily performed.

3. Database Management:

   - Sets play a crucial role in database theory and query languages.

   - SQL, a standard language for relational databases, includes set operations like UNION, INTERSECT, and EXCEPT.

4. Probability and Statistics:

   - Sets are used to model outcomes in probability theory.

   - Events in probability are often represented as sets of possible outcomes.

5. Programming:

   - Sets are employed in algorithms and programming to efficiently handle unique elements and perform set-based operations.

Example:

Consider two sets:

\[ X = \{1, 2, 3, 4, 5\} \]

\[ Y = \{3, 4, 5, 6, 7\} \]

- Union (X ∪ Y): \(\{1, 2, 3, 4, 5, 6, 7\}\)

- Intersection (X ∩ Y):\ \(\{3, 4, 5\}\)

- Difference (X - Y): \(\{1, 2\}\)

These set operations illustrate practical uses in combining, intersecting, or distinguishing elements between sets.