CBSE Class 11 Commerce Maths Sets And Functions Important Topics
CBSE Class 11 Commerce Maths Sets And Functions : 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.
CBSE Class 11 Commerce Maths Sets And Functions Important Topics
CBSE 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}.
CBSE 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.
CBSE 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.
CBSE Class 11 Commerce Maths Sets And Functions Important Topics
CBSE 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 sin^{2}x+cos^{2}x=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:
CBSE 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.
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CBSE Class 11 Commerce Maths Sets And Functions Important Topics
CBSE 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.
For Example
Consider a set A containing elements as {1, 2, 3} and set B contains elements as {2, 4, 6}.
Then the relation between Set A and B from A to B will be set of any combinations from Set A to set B.
http://cdn1.askiitians.com/Images/201739-144035643-9045-1-relations-and-functions.png
From the above diagram, we can see that Relation from A to B i.e. R will be set of {(1,4). (1,2), (3,4), (3,2)}. This relation is a subset of the Cartesian product of two sets A X B.
Let’s take another example where, set A = {1, 2, 3} and set B = {1, 2, 3, 4, 5, 6, 7, 8, 9}.
If the Relation between A and B is as: elements of B is the squares of elements of set A, then the relation is written in the form of sets as:
R = {(a,b): where b is square of a and a ∈ A & b ∈ B}
Then R = {(1,1), (2,4), (3,9)}
Same as sets, relation may also be represented algebraically either by the Roster method or by the Set-builder method.
Relation can also be defined as a linear operation which establishes relationship between the element’s of two set’s according to some definite rule of relationship.
R : {(a, b) | (a, b) ∈ A x B and a R b}
Eg:1 A is {2, 3, 5}
B is {1, 4, 9, 25, 30}
If a R b → b is square of a
Discrete element of relation are {(2, 4), (3, 9). (5, 25)}
Eg: 2 A = {Jaipur, Lucknow, Kanpur, Bhopal}
B = {Rajasthan, Uttar pradesh, Madhya Pradesh}
aRb → a is capital of b,
A X B = {(Jaipur, Rajasthan), (Lucknow, Uttar Pradseh), (Bhopal, Madhya Pradesh}
Total Number of Relation from A to B
Let the number of relations from A to B be x.
Let A contain ‘m’ elements and B contains ‘n’ elements
Number of element’s in A x B → m x n
Number of non void subset’s =
Thus, for A = {1, 2, 3) & B = {x, y, z}
Number of non void subset’s or the number of possible relations = =2^{9 }– 1 = 511
What are Functions?
Functions are the special class of relation or we can say that special types of relations are called as Functions. Function is one of the most important concepts in mathematics as every situation in real life are solved and analysed first by writing its mathematical equation or function.
A function is like a machine which gives unique output for each input that is fed into it. But every machine is designed for certain defined inputs for e.g. a washing machine is designed for washing cloths and not the wood. Similarly the functions are defined for certain inputs which are called http://cdn1.askiitians.com/Images/201739-144045377-1925-2-relations-and-functions.pngas its domain and corresponding outputs are called Range.
CBSE Class 11 Commerce Maths Sets And Functions : Let A and B be two sets and let there exist a rule or manner or correspondence ‘f’ wihch associates to each element of A to a unique element in B, then f is called a Function or Mapping from A to B. It is denoted by symbol
Which reads ‘f is a function from A to B’ or’ f maps A to B.
If an elements a ∈ A is associated with an element b ∈ B then b is called ‘the f image of a’ or ‘image of a under f’ or ‘the value of the function f at a’. Also a is called the pre – image of b or argument of b under the function f. we write it as
f:(a, b) or f:a → b or b = f(a)
A relation f from a set A to a set B is called as the function if it satisfies the below conditions:
- All the elements of A should be mapped with the elements of B. That is, there should not be any element in A which is being unmapped with B.
i.e. ∀a, (a, f(a)) ∈ f, where, a is the elements of set A
- Elements of set A should be uniquely mapped with the elements of set B.
i.e. if (a, b) ∈ f & (a, c) ∈f, ⇒ b = c
Thus the ordered pairs of f must satisfy the property that each element of A appears in some ordered pair and no two ordered pairs have same first elements.
See the below figures to understand the above points.
http://cdn1.askiitians.com/Images/201739-1562863-7704-3-relations-and-functions.png
Note: Every function is a relation but every relation is not necessarily a function.
http://cdn1.askiitians.com/Images/201739-15717893-8642-4-relations-and-functions.jpg
What is the domain of a function? Also explain its co-domain and Range
CBSE Class 11 Commerce Maths Sets And Functions : For a relation from set A to set B i.e. aRb, all the elements of set A are called as the domain of the relation R while all the elements of set B are called as the co-domain of the relation R.
Range is the set of all second elements from the ordered pairs (a, b) in the relation aRb.
Domain of f = {a | a ∈ A, (a, f(a)) ∈ f }
Range of f = {f (a) |a ∈ A, f(a) ∈ B, (a, f (a)) ∈ f}
http://cdn1.askiitians.com/Images/201739-15120122-503-5-relations-and-functions.png
CBSE Class 11 Commerce Maths Sets And Functions : For the relation aRb, domain is considered as the input to relation R while the co-domain is the possible outputs and range is the actual output.
Similarly for a function, f: A → B, elements of set A are the inputs and B is the set of possible outputs. But the second elements of all ordered pair of f(A, B) will be the actual outputs.
It should be noted that range is a subset of co-domain. If only the rule of function is given then the domain of the function is the set of those real numbers, where function is defined. For a continuous function, the interval from minimum to maximum value of a function gives the range.
Which is not the graph of a function?
CBSE Class 11 Commerce Maths Sets And Functions : To identify any graph, weather it is a function or not, we must understand its definition once again but in terms of graphical meaning.
First condition of function says that – “All the elements of A should be mapped with the elements of B”. That means graphically, for every input in its domain, function must give or provide the corresponding output.
Second condition of the function says that – “Elements of set A should be uniquely mapped with the elements of set B”. These means that, for any input x, we must have one and only one output. The best way to check this condition for the function y = f(x), is draw a line parallel to y – axis. If it cuts the graph at two or more distinct points, this means, for one value of x, we are getting more than one outputs. And hence it will not be a function.
Example
y = x^{3}
Here all the straight lines parallel to y – axis cut y = x^{3} only at one point.
http://cdn1.askiitians.com/Images/201739-151536164-6358-6-relations-and-functions.png
Example
x^{2 }+ y^{2 }= 1
Here line parallel to y –axis is intersecting the circle at two points hence it is not a function.
http://cdn1.askiitians.com/Images/201739-151544180-8747-7-relations-and-functions.png
Even Function: Let f(x) be a real valued function of a real variable. Then f is even if the following equation holds for all x and -x in the domain of f:
f(x) = f(-x)
Geometrically, the graph of an even function is symmetric with respect to the y-axis.
Odd Function: Again, let f(x) be a real valued function of a real variable. Then f is odd if the following equation holds for all x and -x in the domain of f:
f (-x) = – f(x) or f(x) + f(-x) = 0.
Increasing function: A function f is said to be increasing if whenever a >b, then
f (a) = f(b). Further a function is said to be strictly increasing if
When a > b, then f(a) > f(b). The graph of an increasing function looks somewhat like this:
http://cdn1.askiitians.com/Images/20131213-92710903-5013-img.gif
You may also refer the video on increasing function
What are the Classification of functions?
Functions are classified as follows:
- Polynomial Function: If a function f is defined by
CBSE Class 11 Commerce Maths Sets And Functions : where n is a non negative integer and a_{n}, a_{n-1}, …, a, a_{0} are real number and a_{n} ≠ 0, then f is called a Polynomial Function of degree n. A polynomial function is always continuous.
- Algebraic Function: A function f is called an algebraic function if it can be constructed using algebraic operations such as addition, subtraction, multiplication, division and taking roots, started with polynomials.
Example
Note: All the polynomials are algebraic but converse is not true. Functions which are not algebraic, are known as Transcidental Function.
- Fractional Rational Function: A rational function is a function of the form, are polynomials & h(x) ≠ 0. The domain of f(x) is set of real x such that h(x) ≠ 0.
Example
- Exponential Function: A function f(x) = a^{x }= e^{xlna}(a > 0, a ≠ 1, x ∈ R) is called an Exponential Function. f(x) = a^{x} is called an exponential function because the variable x is the exponent. It should not be confused with power function. g(x) = x^{2} in which variable x is the base. For f(x)= e^{x }domain in R and range is R^{+}.
- Logarithmic Function: A function of the form y=log_{a} x, x > 0, a > 0, a ≠ 1, is is called Logarithmic Function.
- Absolute Value Function (or Modulus Function): A function y= f(x) = |x| is called the Absolute Value Function or Modulus Function. It is defined as:
For f(x) = |x|,domain is R and Range is R^{+} U {0}.See below for its figure.
http://cdn1.askiitians.com/Images/201739-1624657-6244-10-relations-and-functions.png
- http://cdn1.askiitians.com/Images/201739-16215970-2996-11-relations-and-functions.pngSignum Function: A function y= f(x) = sgn (x) is defined as follows:
It is also written as
- Greatest Integer Or Step up Function: The function y = f(x) = [x] is called the greatest integer function where [x] denotes the greatest integer less than or equal to x. Note that for:
http://cdn1.askiitians.com/Images/201739-16232438-1221-12-relations-and-functions.png
-1 ≤ x <0; [x] = -1
0 ≤ x <1; [x] = 0
1 ≤ x < 2; [x] = 1
2 ≤ x < 3; [x] = 2 and so on.
For f(x) = [x], domain is R and range is I.
- Identity Function: The f: A → B defined by f(x) = x, ∀ x ∈A is called the identity of A and is denoted by I_{A. }The domain and range of identity function is entire real range i.e. R f(x) = x
http://cdn1.askiitians.com/Images/201739-16242485-9292-13-relations-and-functions.png
- http://cdn1.askiitians.com/Images/201739-16251672-4506-14-relations-and-functions.png
- Constant Function: The function f: A → B is said to be a constant function if every element of A has the same f image in B. Thus f : A → B; f(x) = c,∀ x ∈ A, c ∈ B is a constant function. Note that the range of a constant function is a singleton and a constant function may be one – one or many – one, onto or into.
Example
f(x) = [{x}]; g(x) =sin^{2}x + cos^{2}x; h (x) = sgn (x^{2}– 3x + 4) etc, all are constant functions.
CBSE Class 11 Commerce Maths Sets And Functions Important Topics
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