JKBOSE Class 11th Mathamatics Notes PDF Download
JKBOSE Class 11th Mathamatics Notes PDF Download
JKBOSE Class 11th Mathamatics Notes PDF Download. Get the important of Class 11th for the subject Mathamatics. All the Units of this subject is discussed below unitwise. You can choose the unit below that you want. In this article you will get complate JKBOSE Class 11th Mathamatics notes with PDF Download.
Download UnitWise JKBOSE Class 11th Mathamatics PDF Notes
‣ Sets and their Representations
In mathematics, a set is denoted by enlisting its constituents within curly braces. Let us consider an instance of a set named A, comprising the elements 1, 2, and 3. We denote this set as A = {1, 2, 3}. Remember, the arrangement of elements is inconsequential, and duplicates are not admissible within a set. Sets can encompass diverse entities, including numbers, letters, or even other sets.
➜ Empty Set: The Foundational Core of SetsConceptualize an empty set as an extraordinary entity bereft of elements. We symbolize it using the notations ∅ or {}. It serves as the genesis of sets and assumes immense significance in the realm of mathematics. An empty set is employed when we wish to represent a collection devoid of any constituents. For instance, suppose we have a set named B = {x | x is a prime number greater than 10 and less than 20}. Upon scrutiny, we ascertain that B is an empty set, for no prime numbers fulfill those conditions.➜ Finite and Infinite Sets: Enumerating the Elements Sets can be classified into two categories: finite and infinite. A finite set encompasses a fixed number of elements. For example, let us consider the set C = {1, 2, 3, 4, 5}. It is a finite set comprising five elements. Conversely, an infinite set boasts an immeasurable multitude of elements. One such example is the set of natural numbers, N = {1, 2, 3, 4, ...}, an unending sequence that stretches into perpetuity.➜ Equivalent Sets: When two sets possess identical elements, they are deemed equivalent. The order of elements holds no significance. Therefore, if we have two sets, D = {1, 2, 3} and E = {3, 2, 1}, we can declare that D = E owing to their shared constituents.➜ Subsets: A subset denotes a set that encompasses all the elements of another set. In simpler terms, if all the constituents of set F are also found in set G, then F is considered a subset of G. This relationship is represented as F ⊆ G. Furthermore, it is worth noting that every set is always deemed a subset of itself.
Let us consider an illustration: we possess sets H = {1, 2} and I = {1, 2, 3}. In this scenario, H constitutes a subset of I since all the elements of H are also present in I. This relationship can be depicted as H ⊆ I.
‣ Relations and Functions
Relations and functions are concepts that establish connections among elements derived from distinct sets.
➜ Ordered Pairs: An ordered pair refers to a unique pairing of elements arranged in a specific order. Unlike sets, the sequence of elements is consequential in this context. Ordered pairs are frequently utilized to demonstrate relationships between elements sourced from different sets.
For instance, let us consider sets J = {1, 2, 3} and K = {a, b, c}. By combining an element from set J with an element from set K, we can create ordered pairs such as (1, a), (1, b), (1, c), (2, a), (2, b), (2, c), (3, a), (3, b), and (3, c).
➜ Cartesian Product of Sets: The Cartesian product of two sets, denoted as A × B, entails combining their constituents to form pairs. In essence, it serves as a means to connect elements from two sets and generate ordered pairs.
For example, let us consider set L = {1, 2} and set M = {x, y}. The Cartesian product L × M would yield {(1, x), (1, y), (2, x), (2, y)}.
➜ Number of Elements in the Cartesian Product: To determine the number of elements in the Cartesian product of two finite sets, we multiply the number of elements in the first set by the number of elements in the second set. Hence, if set A comprises m elements and set B comprises n elements, the Cartesian product A × B will encompass m × n elements.
For instance, let us consider two finite sets P = {1, 2, 3} and Q = {a, b}. The Cartesian product P × Q will yield a total of 3 × 2 = 6 elements.
FAQs on JKBOSE Class 11th Mathamatics Subject
➥ Why does the empty set hold significance in set theory?
The empty set represents the cornerstone of sets. It embodies a collection devoid of elements and assumes a pivotal role in mathematical proofs. The empty set facilitates the definition of concepts such as subsets and set operations.
➥ How can one ascertain if two sets are equal?
To determine the equality of two sets, one must verify whether they possess precisely the same elements. The order of elements is inconsequential. If every element of set A is also an element of set B, and vice versa, then sets A and B are considered equal.
➥ What distinguishes finite sets from infinite sets?
A finite set comprises a finite number of elements, whereas an infinite set encompasses an uncountable multitude of elements. Finite sets possess an endpoint, whereas infinite sets persist indefinitely.
➥ How can one calculate the number of elements in the Cartesian product of two sets?
To determine the number of elements in the Cartesian product of two finite sets, one must multiply the number of elements in the first set by the number of elements in the second set. This computation yields the total count of elements in the Cartesian product.
➥ Why are ordered pairs employed in mathematics?
Ordered pairs aid in the representation of relationships between elements sourced from distinct sets. They enable the establishment of connections and facilitate the description of mappings between diverse mathematical entities.
➥ What is a subset?
A subset refers to a set that encompasses all the elements of another set. If all the elements of set A are also present in set B, then set A is considered a subset of set B.
‣ Conclusion
Sets and their representations serve as the fundamental building blocks of modern mathematics. They enable us to grasp complex problems and enhance our understanding. By acquiring knowledge about concepts such as empty sets, finite and infinite sets, equivalent sets, subsets, relations, functions, ordered pairs, the Cartesian product, and the cardinality of the Cartesian product of two finite sets, we establish a solid foundation
