Important Questions and Solutions on Sets, Relations, and Functions
FAQs on Sets, Relations, and Functions Practice Paper for Students
1. What are the basic concepts of sets, relations and functions?
Sets, relations, and functions form the foundation of modern mathematics and are crucial chapters in the CBSE syllabus.
- Sets: A collection of well-defined objects.
- Relations: A way to describe connections between elements of two sets.
- Functions: A special relation where each element of the first set is linked with exactly one element of the second set.
2. Define a set and explain its types with examples.
A set is a collection of distinct, well-defined objects, called elements.
- Finite Set: Has a countable number of elements (e.g., {1,2,3}).
- Infinite Set: Elements cannot be counted (e.g., set of all natural numbers).
- Empty Set: Contains no elements (e.g., { }).
- Singleton Set: Contains only one element (e.g., {5}).
- Universal Set: Contains all objects under consideration, usually denoted by U.
3. What is the difference between a relation and a function?
A relation is a connection between elements of two sets, while a function is a specific type of relation.
- Relation: Can pair one element of set A with multiple elements of set B.
- Function: Each element of set A pairs with exactly one element of set B.
4. How do you represent a set? List three methods.
There are three main ways to represent a set:
- Roster Method: List all elements (e.g., A = {2, 4, 6}).
- Set-builder Method: State a property all members share (e.g., A = {x : x is an even natural number}).
- Venn Diagram: Pictorial representation using circles.
5. State and explain properties of operations on sets.
Operations on sets have several properties:
- Union: A ∪ B contains all elements from A, B, or both.
- Intersection: A ∩ B contains only elements common to both A and B.
- Difference: A – B has elements in A but not in B.
- Complement: The set of all elements not in a set.
- Demorgan's Laws: Relate union and intersection of complements.
6. What is an equivalence relation? Give an example.
An equivalence relation is a relation that is reflexive, symmetric, and transitive.
- Reflexive: Every element is related to itself.
- Symmetric: If a is related to b, then b is related to a.
- Transitive: If a is related to b and b to c, then a is related to c.
7. How can you determine if a relation is a function?
A relation is a function if every input has exactly one output.
- Each element of the domain must be paired with only one element in the codomain.
- The vertical line test (in graphs) helps verify this property.
8. What are one-one and onto functions? Explain with examples.
One-one functions (Injective) map each element of the domain to a unique element of the codomain, while onto functions (Surjective) ensure every element of the codomain is covered.
- One-one Example: f(x) = 2x (distinct inputs give distinct outputs).
- Onto Example: f(x) = x^3 (all real numbers are possible outputs).
9. What is the Cartesian product of two sets? Give an example.
The Cartesian product of two sets A and B, written as A × B, is the set of all ordered pairs (a, b) where a ∈ A and b ∈ B.
- Example: If A = {1,2}, B = {x,y}, then A × B = {(1, x), (1, y), (2, x), (2, y)}.
10. What are the applications of sets, relations and functions in real life?
Sets, relations, and functions have wide applications in everyday life and technology.
- Computer databases: Storing and relating data items.
- Social networks: Modeling relationships and connections between people.
- Functions in science: Expressing physical laws and mathematical models.
- Classification: Grouping objects based on properties (sets).
