**KARNATAKA class 11 commerce Maths fast track revision notes**

KARNATAKA class 11 commerce Maths fast track revision notes : The Central Board of Secondary Education (abbreviated as CBSE) is a Board of Education for public and private schools, under the Union Government of India. Central Board of Secondary Education (CBSE) has asked all schools affiliated to follow only NCERT curriculum.

**KARNATAKA class 11 commerce Maths fast track revision notes**

Class 11 Commerce Maths Sets And Functions : A set is an underscored collection of distinct objects. We use the notation x ∈ S to mean “x is an element of S” and x /∈ S to mean “x is not an element of S.” Given two subsets (sub collections) of U, X and Y , we say “X is a subset of Y ,” written X ⊆ Y , if x ∈ X implies that x ∈ Y . Alternatively, we may say that “Y is a super set of X.” X ⊆ Y and Y ⊇ X mean the same thing. We say that two subsets X and Y of U are equal if X ⊆ Y and Y ⊆ X. We use braces to designate sets when we wish to specify or describe them in terms of their elements: A = {a, b, c}, B = {2, 4, 6, . . .}. A set with k elements is called a k-set or set with carnality k. The cardinality of a set A is denoted by |A|.

Since a set is an unordered collection of distinct objects, the following all describe the same 3-element set

{a, b, c} = {b, a, c} = {c, b, a} = {a, b, b, c, b}.

Class 11 Commerce Maths Sets And Functions : The first three are simply listing the elements in a different order. The last happens to mention some elements more than once. But, since a set consists of distinct objects, the elements of the set are still just a, b, c. Another way to think of this is:

Two sets A and B are equal if and only if every element of A is an element of B and every element of B is an element of A.

Class 11 Commerce Maths Sets And Functions : Thus, with A = {a, b, c} and B = {a, b, b, c, b}, we can see that everything in A is in B and everything in B is in A. You might think “When we write a set, the elements are in the order written, so why do you say a set is not ordered?” When we write something down we’re stuck — we have to list them in some order. You can think of a set differently: Write each element on a separate slip of paper and put the slips in a paper bag. No matter how you shake the bag, it’s still the same set.

**KARNATAKA class 11 commerce Maths fast track revision notes**

Class 11 Commerce Maths Sets And Functions : The basic concepts of sets and functions are topics covered in high school math courses and are thus familiar to most university students. We take the intuitive point of view that sets are underscored collections of objects. We first recall some standard terminology and notation associated with sets. When we speak about sets, we usually have a “universal set” U in mind, to which the various sets of our discourse belong.

1. Sets

Sets and their representations. Empty set. Finite and Infinite sets. Equal sets. Subsets. Subsets of a set of real numbers especially intervals (with notations). Power set. Universal set. Venn diagrams. Union and Intersection of sets. Difference of sets. Complement of a set. Properties of Complement Sets. Practical Problems based on sets.

2. Relations & Functions

Ordered pairs, Cartesian product of sets. Number of elements in the cartesian product of two finite sets. Cartesian product of the sets of real (upto R x R). Definition of relation, pictorial diagrams, domain, co-domain and range of a relation. Function as a special kind of relation from one set to another. Pictorial representation of a function, domain, co-domain and range of a function. Real valued functions, domain and range of these functions: constant, identity, polynomial, rational, modulus, signum, exponential, logarithmic and greatest integer functions, with their graphs. Sum, difference, product and quotients of functions.

3. Trigonometric Functions

Positive and negative angles. Measuring angles in radians and in degrees and conversion of one into other. Definition of trigonometric functions with the help of unit circle. Truth of the sin2x+cos2x=1, for all x. Signs of trigonometric functions. Domain and range of trignometric functions and their graphs. Expressing sin (x±y) and cos (x±y) in terms of sinx, siny, cosx & cosy and their simple application. Deducing identities like the following:

Class 11 Commerce Maths Sets And Functions : Identities related to sin 2x, cos 2x, tan 2x, sin 3x, cos 3x and tan 3x. General solution of trigonometric equations of the type sin y = sin a, cos y = cos a and tan y = tan a.

Download here CBSE Class 11 Commerce Maths Sets And Functions notes in PDF Format

**KARNATAKA class 11 commerce Maths fast track revision notes**

Class 11 Commerce Maths Sets And Functions : This topic “Relations and Functions” is a foundation or fundamental of algebra in mathematics. Relations and functions are two different words having different meaning mathematically. Many us you might be confused in their difference. We shall study both these concepts in detail here.

Same as the relations which we have in our daily life, a kind of relations also exists in algebra. In daily life, relations are like brother and sister, friends, student and teacher and many more. In mathematics also we see some relations like a line is parallel or perpendicular to another, any one variable is greater or less than the another variable. Any Set A is subset of B, all these are examples of relations.

One thing which we see in common while studying relations, that it required two different objects to link two different objects via relations.

**What is the meaning of Relation in math?**

Understanding Relations requires basic knowledge of sets. A Set is a collection of well defined objects of particular kind. For Example a set of outcomes of dice, a set of English alphabet.

Relation is always studied between two sets. If we have two non void (or null/empty) sets A and B then the relation R from set A to set B is represented by aRb, where a is the set of elements belonging to set A while b belongs to set B.

Relation from a set A to a set B is the subset of the Cartesian product of A and B i.e. subset of A x B. Relation in other way can also be defined as an collection of ordered pairs (a, b) where a belongs to the elements from set A and b from set B and the relation is from A to B but not vice versa.

**KARNATAKA class 11 commerce Maths fast track revision notes**

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