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Relation and Function Class 12 Maths | NCERT Chapter 1 | CBSE JEE | One Shot |हिंदी में
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Overview
This video provides a comprehensive one-shot lesson on Relations and Functions for Class 12 Mathematics, covering essential concepts from NCERT. It begins by defining relations in the context of sets and real-world examples, then delves into types of relations: empty and universal. The core of the video focuses on different types of relations: reflexive, symmetric, and transitive, illustrating each with examples in both roster and set-builder forms. It also introduces equivalence relations and explores functions, including one-to-one (injective), onto (surjective), and bijective functions, with proofs and examples. Finally, the video touches upon the composition of functions, invertible functions, and binary operations, including their properties like commutative, associative, identity, and inverse elements, using various number sets like integers and real numbers.
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Chapters
- Relations exist between elements of sets, similar to real-world relationships.
- A relation R from set A to set B is a subset of the Cartesian product A x B.
- Empty relation: A relation with no elements from the set. Example: Difference between ages of students from nursery to 12th grade is more than 100 years.
- Universal relation: A relation true for all elements of the set. Example: Difference between ages of any two students is less than 100 years.
- Relations can be denoted as (a, b) ∈ R or a R b.
Understanding the fundamental definition and types of relations is crucial for building a strong foundation in set theory and preparing for more complex mathematical concepts.
The difference between the ages of students in a school being greater than 100 years (empty relation) or less than 100 years (universal relation).
- Reflexive relation: For every element 'a' in set A, (a, a) must belong to the relation R.
- Symmetric relation: If (a, b) belongs to R, then (b, a) must also belong to R.
- Transitive relation: If (a, b) belongs to R and (b, c) belongs to R, then (a, c) must also belong to R.
- Equivalence relation: A relation that is reflexive, symmetric, and transitive.
These properties define the structure and behavior of relationships between elements, which are fundamental for classifying relations and solving problems involving them, especially for proving equivalence relations.
For a set of real numbers, the relation 'a ≤ b' is reflexive, symmetric (if a ≤ b and b ≤ a, then a=b), and transitive, making it an equivalence relation.
- A function is a special type of relation where each element in the domain maps to exactly one element in the codomain.
- Injective (One-to-one) function: Distinct elements in the domain map to distinct elements in the codomain.
- Surjective (Onto) function: Every element in the codomain is mapped to by at least one element in the domain (no element in the codomain is left unpaired).
- Bijective function: A function that is both injective and surjective.
Functions are the building blocks of many mathematical models and real-world applications, and understanding their properties (like being one-to-one or onto) is key to analyzing their behavior and applicability.
The function f(x) = x² from integers to integers is not surjective because negative integers in the codomain have no corresponding integer in the domain.
- Composition of functions (g ∘ f) means applying one function after another.
- If f: X → Y and g: Y → Z, then g ∘ f: X → Z.
- A function is invertible if and only if it is bijective (both one-to-one and onto).
- The inverse function 'f⁻¹' reverses the mapping of the original function 'f'.
Function composition allows us to combine multiple operations, and understanding inverse functions is crucial for solving equations and understanding transformations.
If f(x) = 2x and g(x) = x+1, then (g ∘ f)(x) = g(f(x)) = g(2x) = 2x + 1.
- A binary operation '*' on a set S is a function that takes two elements from S and returns a single element in S.
- Closure property: For any a, b ∈ S, a * b ∈ S.
- Commutative property: For any a, b ∈ S, a * b = b * a.
- Associative property: For any a, b, c ∈ S, (a * b) * c = a * (b * c).
- Identity element: An element 'e' ∈ S such that a * e = e * a = a for all a ∈ S.
- Inverse element: For an element 'a' ∈ S, there exists an element 'a⁻¹' ∈ S such that a * a⁻¹ = a⁻¹ * a = e (where 'e' is the identity element).
Binary operations and their properties (like commutative, associative, identity, inverse) are fundamental to abstract algebra and are used to define algebraic structures like groups, rings, and fields.
Addition of real numbers is a binary operation. It is commutative (a+b = b+a), associative ((a+b)+c = a+(b+c)), has an identity element (0), and each element 'a' has an additive inverse (-a).
Key takeaways
- Relations are defined on sets, and their properties (reflexive, symmetric, transitive) determine their structure.
- Functions are special relations where each input has exactly one output, with key types being injective, surjective, and bijective.
- Composition of functions allows chaining operations, while inverse functions reverse them, requiring the function to be bijective.
- Binary operations combine two elements of a set to produce a result within the same set, and they can possess properties like closure, commutativity, associativity, and have identity and inverse elements.
- Understanding these concepts is crucial for advanced mathematics, including abstract algebra and calculus.
Key terms
RelationCartesian ProductReflexive RelationSymmetric RelationTransitive RelationEquivalence RelationFunctionInjective FunctionSurjective FunctionBijective FunctionComposition of FunctionsInverse FunctionBinary OperationCommutative PropertyAssociative PropertyIdentity ElementInverse Element
Test your understanding
- How do the properties of reflexive, symmetric, and transitive relations help classify different types of relationships between sets?
- What is the difference between an injective and a surjective function, and why is bijectivity necessary for a function to have an inverse?
- How does the closure property define a binary operation, and what are the implications if an operation does not satisfy closure on a given set?
- Explain the role of identity and inverse elements in the context of binary operations and how they relate to algebraic structures.
- Given a relation, what steps would you take to determine if it is reflexive, symmetric, and transitive?