Elements of Mathematics Class 11 Solutions Chapter 5: A Comprehensive Guide

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Chapter 5 of Class 11 Mathematics, often referred to as "Elements of Mathematics," introduces students to fundamental concepts like sets, relations, and functions. These are crucial topics in mathematics that form the foundation for various advanced concepts in the subject. The chapter begins by explaining what sets are and how they can be represented, along with their different types such as finite, infinite, equal, and null sets. Students are taught how to work with these sets, perform operations like union, intersection, and complement, and understand their properties in various contexts.

Question and Answer Section

Question:
What are the basic concepts of Chapter 5 in Class 11 Mathematics?

Answer:
Chapter 5 covers the foundational concepts of sets, relations, and functions. It explores different types of sets and their properties, the concept of relations between sets, and the importance of functions in mathematics.

Question:
What is a set in mathematics?

Answer:
A set is a collection of distinct objects, considered as an object in its own right. It is represented by curly braces, and the elements inside the set are separated by commas.

Question:
How is a set represented in mathematics?

Answer:
A set is typically represented using curly brackets, such as {1, 2, 3, 4}, where 1, 2, 3, 4 are elements of the set.

Question:
What are the types of sets?

Answer:
There are various types of sets, such as finite sets, infinite sets, equal sets, null sets, singleton sets, and universal sets.

Question:
What is the power set of a given set?

Answer:
The power set of a set is the set of all possible subsets of that set, including the empty set and the set itself.

Question:
Define the union of two sets.

Answer:
The union of two sets A and B is the set of elements that are in A, in B, or in both. It is represented by A ∪ B.

Question:
What is an intersection of sets?

Answer:
The intersection of two sets A and B is the set of elements that are common to both sets. It is represented by A ∩ B.

Question:
What is a null set?

Answer:
A null set, also called an empty set, is a set that does not contain any elements. It is denoted by {} or Ø.

Question:
Explain the concept of a universal set.

Answer:
The universal set is the set that contains all the elements under consideration for a particular discussion. It is usually denoted by 'U'.

Question:
What is a subset in mathematics?

Answer:
A set A is considered a subset of set B if every element of A is also an element of B. This is denoted as A ⊆ B.

Question:
What is the complement of a set?

Answer:
The complement of a set A refers to all the elements that are in the universal set but not in A. It is denoted by A'.

Question:
What are relations in mathematics?

Answer:
A relation between two sets is a collection of ordered pairs where the first element is from the first set and the second element is from the second set.

Question:
How are relations represented?

Answer:
Relations are typically represented as sets of ordered pairs, such as {(1, 2), (2, 3), (3, 4)}.

Question:
What is a function in mathematics?

Answer:
A function is a special type of relation where each element of the first set is related to exactly one element of the second set.

Question:
What are the types of functions?

Answer:
Functions can be classified as one-to-one, onto, many-to-one, and many-to-many, based on how the elements of the sets are mapped.

Question:
What is an identity function?

Answer:
An identity function is a function where every element of the set is mapped to itself. It is represented by f(x) = x.

Question:
What is a constant function?

Answer:
A constant function is a function where every element in the domain is mapped to the same element in the codomain. It is represented as f(x) = c, where c is a constant.

Question:
What is an injective function?

Answer:
An injective function, or one-to-one function, is a function where different elements in the domain map to different elements in the codomain.

Question:
What is a surjective function?

Answer:
A surjective function, or onto function, is a function where every element of the codomain is related to at least one element in the domain.

Question:
What is a bijective function?

Answer:
A bijective function is both injective and surjective, meaning every element of the domain maps to a unique element of the codomain, and every element of the codomain has a corresponding element in the domain.

Question:
What is the composition of functions?

Answer:
The composition of two functions f and g is a function that applies g first and then f to the result, denoted as (f ∘ g)(x) = f(g(x)).

Question:
What is an inverse function?

Answer:
An inverse function is a function that reverses the effect of the original function. If f(x) maps x to y, then the inverse function f⁻¹(y) maps y back to x.

Question:
How can you find the inverse of a function?

Answer:
To find the inverse of a function, swap the roles of x and y in the equation and solve for y. The resulting equation gives the inverse function.

Question:
What is a Cartesian product of two sets?

Answer:
The Cartesian product of two sets A and B, denoted by A × B, is the set of all ordered pairs (a, b) where a is an element of A and b is an element of B.

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