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Chapter Analysis
Advanced16 pages • EnglishQuick Summary
This chapter introduces the concept of relations and functions, expanding upon the knowledge from Class XI. It delves into different types of relations, functions, including their compositions and invertibility, along with binary operations. The chapter focuses on special kinds of relations like equivalence relations and explorations into injective, surjective, and bijective functions.
Key Topics
- •Relations
- •Functions
- •Composition of functions
- •Invertible functions
- •Binary operations
- •Equivalence relations
- •One-one and onto functions
Learning Objectives
- ✓Understand different types of relations.
- ✓Analyze functions and their compositions.
- ✓Evaluate the invertibility of functions.
- ✓Discern binary operations in the context of functions.
- ✓Identify and explore equivalence relations.
- ✓Characterize functions as injective, surjective, and bijective.
Questions in Chapter
Determine whether each of the following relations are reflexive, symmetric and transitive.
Page 6
Show that the relation R in the set R of real numbers, defined as R = {(a, b) : a ≤ b2} is neither reflexive nor symmetric nor transitive.
Page 6
Check whether the relation R defined in the set {1, 2, 3, 4, 5, 6} as R = {(a, b) : b = a + 1} is reflexive, symmetric or transitive.
Page 6
Show that the relation R in R defined as R = {(a, b) : a ≤ b}, is reflexive and transitive but not symmetric.
Page 6
Show that the function f : R* → R* defined by f(x) = 1/x is one-one and onto.
Page 11
Check the injectivity and surjectivity of the following functions: f : N → N given by f(x) = x², f : Z → Z given by f(x) = x².
Page 11
Additional Practice Questions
Explain the concept of equivalence relations with examples.
mediumAnswer: Equivalence relations are those that are reflexive, symmetric, and transitive. For example, the 'congruent to' relation in geometry, where every shape is congruent to itself (reflexive), if one shape is congruent to another, then the second is congruent to the first (symmetric), and if one shape is congruent to a second, and that second is congruent to a third, then the first is congruent to the third (transitive).
Create an example to illustrate the composition of two functions.
mediumAnswer: Consider two functions f(x) = 2x and g(x) = x + 3. The composition g(f(x)) means substituting f(x) into g, leading to g(2x) = 2x + 3. Hence, g(f(x)) = 2x + 3.
Differentiate between injective, surjective, and bijective functions with examples.
hardAnswer: An injective function assigns distinct outputs to distinct inputs, e.g., f(x) = 2x. A surjective function covers the entire range, e.g., g(x) = x² for all real x. A bijective function is both injective and surjective, e.g., f(x) = x is a bijective function from real numbers to real numbers.