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Relation And Function Class 12 | Class 12 Full Chapter 1 | Important Questions/One Shot/All Exercise

Dear Sir・116 minutes read

The text discusses relationships within sets, focusing on reflexivity and transitivity, as well as functions like injective relationships. It explains the importance of different types of functions and their application in understanding relationships between numbers.

Insights

  • Relationships and functions are distinct concepts, with relationships focusing on connections between individuals under specific conditions, while functions involve mapping elements from one set to another.
  • Different types of relationships, such as symmetrical, transitive, and equivalence relations, are crucial in understanding how elements interact based on conditions like reflexivity, symmetry, and transitivity.
  • Functions play a vital role in mapping elements between sets, with one-to-one, on-to, and bijective functions ensuring unique pre-images for each element, highlighting the importance of injectivity and subjectivity in defining relationships accurately.

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Recent questions

  • What is a relationship?

    A connection between individuals with specific conditions.

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Summary

00:00

Understanding Relationships and Functions in Mathematics

  • Relationship and function are discussed as two separate concepts.
  • Relationship is defined as a connection between at least two individuals, with specific conditions.
  • The concept of Cartesian product is introduced using sets A and B with elements 1, 2, 3 and 4, 5, 6 respectively.
  • The importance of conditions in relationships is highlighted, with examples of forming relationships based on conditions.
  • The concept of MT relation is explained, where elements must meet a specific condition, such as multiplication equaling 20.
  • Universal relation is discussed as a type of MT relation where all elements are included.
  • Symmetrical relation is explained as a relationship where elements are reversed in order.
  • The example of a symmetrical relation is demonstrated using mirror symmetry in shapes.
  • The definition and characteristics of symmetrical relations are detailed, emphasizing the reverse order of elements.
  • Transitive relation is introduced as a type of relationship where if a1 is related to a2 and a2 is related to a3, then a1 is also related to a3.

13:30

"Sequencing Elements: Understanding Relationships in Mathematics"

  • The product of 1, 2, and 3 is the starting point.
  • Elements are introduced based on a specific relation forming a sequence.
  • The elements include 1, 2, 2, 3, and 1, denoted as a1, a2, and a3.
  • The common element is identified and removed to isolate the remaining elements.
  • The relationship is analyzed for transitivity, with a1, a2, and a3 being crucial.
  • The exercise focuses on understanding and applying different types of relationships.
  • Steps are provided to establish relationships based on given conditions.
  • A specific condition involves elements being divisible by each other.
  • The process involves multiplying elements within sets to determine relationships.
  • The analysis includes checking for reflexivity, symmetry, and transitivity in the relationships formed.

27:31

Understanding Relationships in Sets: Reflexivity, Symmetry, Transitivity

  • The text discusses the concept of relationships within a set, focusing on reflexivity and transitivity.
  • Reflexivity is explained as a condition where a relationship is defined within a set.
  • Symmetry is introduced as a concept that needs to be proven within a relationship.
  • The text emphasizes the importance of practice to understand and master these concepts.
  • Real numbers are discussed, including positive, negative, rational, irrational, and divisible numbers.
  • The conditions for proving reflexivity within a relationship are detailed, involving comparisons between elements.
  • Symmetry is explored through examples of pairs of numbers and the conditions they must meet.
  • The concept of transitivity is introduced, requiring proof within a relationship.
  • Equivalence relations are explained as relationships that are reflexive, symmetric, and transitive.
  • A practical example is provided to demonstrate how to prove a relationship's equivalence by checking the evenness of the modulus of differences between elements.

40:07

"Numbers, Relationships, and Symmetry: A Mathematical Exploration"

  • The text discusses relationships and symmetry, emphasizing the importance of reflexive relationships.
  • It explains the concept of symmetric relationships, highlighting the relationship between a1 and a2.
  • The text delves into the significance of numbers and their relationships, such as 1 and 3 leading to 31.
  • It introduces the idea of even numbers and their role in relationships, with examples like 1 and 3 forming 31.
  • The text progresses to discuss relationships between numbers like 2 and 4 leading to 42.
  • It explores the concept of transitivity in relationships, showcasing how numbers like 1, 3, and 5 are interconnected.
  • The text transitions to discussing equivalence relations and the proof of elements related to each other.
  • It elaborates on the process of proving relationships between elements, focusing on differences and even numbers.
  • The text then shifts to explaining functions and the importance of non-repeating domains in defining functions.
  • It concludes by detailing different types of functions, including injective functions that have a one-to-one relationship between elements.

53:45

Understanding Functions and Relationships in Mathematics

  • The relationship or function may or may not have been formed between elements of set b or set y.
  • If no function has been established, individuals will be counted based on the domain.
  • The range now includes a, b, c, and d within set a.
  • The domain's meaning is crucial in determining the relationship.
  • The image of b will become c, with a being the first image, b the second, and so forth.
  • The range is always less than or equal to the domain, ensuring the range is within the domain.
  • The function for x will equal the function for y if x and y are equal.
  • Examples are used to illustrate the function's uniqueness and equality.
  • Different types of functions are explained: one-one, many-one, on-to, and bijective.
  • Practical examples are provided to demonstrate the application of these function types.

01:06:48

Exploring Functions and Real Numbers Relationships

  • Real numbers encompass all numbers, including even, odd, positive, negative, rational, and irrational numbers, except zero.
  • Functions are crucial in understanding relationships between numbers and their inverses.
  • Reversing a function involves reciprocating it, such as turning 1/x into x.
  • The concept of one-to-one functions is explored, where each element has a unique pre-image.
  • The process of proving one-to-one functions involves setting conditions and verifying equality between elements.
  • The on-to function ensures that every element has a pre-image within the set.
  • Transitioning from real numbers to natural numbers alters the function's behavior, particularly with negative numbers.
  • Injectivity and subjectivity are essential to check in functions, ensuring each element has a unique pre-image.
  • The function of squaring numbers and its inverse, square root, impacts the relationship between elements.
  • Understanding the behavior of functions with natural numbers aids in determining if a function is on-to or not.

01:19:57

Functions and Integers: Properties and Applications

  • x1 and x2 are natural numbers, and if a function is applied to x1 and x2, the result is x1 squared and x2 squared respectively.
  • The condition for a function to be one-to-one is that the function for x1 must be equal to the function for x2.
  • If the condition for a one-to-one function is met, the squares of x1 and x2 cancel each other out.
  • The function must have a pre-image for every element in order to be onto.
  • If the cube root of a positive integer is taken, the result may or may not be a natural number.
  • The function f(x) = x^3 is injective as it has a unique output for each input.
  • The function f(x) = x^3 is not subjective as not every element has a pre-image.
  • The greatest integer function is defined as the greatest integer less than or equal to x.
  • The greatest integer function maps real numbers to real numbers.
  • The greatest integer function provides the largest integer less than or equal to a given real number.

01:32:38

"Functions and Modulus: Exploring Mathematical Relationships"

  • The number 2.3 will be deducted if 2.7 is taken.
  • The concept of the greatest integer is discussed.
  • The function goes from real numbers to real numbers.
  • The function is explained for x1 and x2.
  • The function is not one-to-one due to multiple elements having the same image.
  • The modulus function is discussed, emphasizing that it always gives a positive value.
  • The proof is provided that the modulus function is not one-to-one.
  • The sign function is explained, where positive inputs yield a positive output and negative inputs yield a negative output.
  • The sign function is illustrated with examples of positive and negative inputs.
  • The sign function is proven to not be one-to-one due to multiple elements having the same image.

01:45:13

"Function y = f(x) creates unique images"

  • A bean image is created when the bean image is formed.
  • The function for this is written as y = f(x).
  • If y is replaced with th, no number results in 3.
  • The result will be positive, zero, or negative.
  • The edge y = 3 does not create an image.
  • The function from set A to set B is given.
  • To prove the function is one-one, each element must have a unique image.
  • The function changes the position of elements from A to B.
  • The function is proven to be unique and one-one.
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