Product mathematically signifies the result obtained when two or more values are multiplied together. For example, 45 is the product of 9 and 5. One must be familiar with the basic operations on sets like Union and Intersection, which are performed on 2 or more sets.
Cartesian Product is also one such operation that is performed on two sets, which returns a set of ordered pairs.
In this article, we have covered, the ordered pair definition, a cartesian product of sets, and others in detail.
Table of Contents
What is an Ordered Pair?
An ordered pair has two parts. The first part is called the first component. The second part is called the second component. We write an ordered pair like this: (a, b). The letter ‘a’ is the first component. The letter ‘b’ is the second component. An ordered pair has two things. One thing comes first. The other thing comes second.
Example:
(5, 7) is an ordered pair of integers.
Note: (5, 7) ≠ (7, 5), an ordered pair (a, b) is equal to (x, y) only if a = x and b = y.
Cartesian Product of Sets
When two sets have items in them, A and B, their Cartesian product is all the pairs you can make. One part of the pair comes from set A. The other part comes from set B. We make every possible pair this way. The result is a new set of all these pairs. We write this new set as A×B.
A × B = {(a, b) : a ∈ A and b ∈ B}
Example:
Let A = {1, 2} and B = {4, 5, 6}
A × B = {(1, 4), (1, 5), (1, 6), (2, 4), (2, 5), (2, 6)}
Here the first component of every ordered pair is from set A the second component is from set B.
Cartesian Product of two sets can be easily represented in the form of a matrix where both sets are on either axis, as shown in the image below. Cartesian Product of A = {1, 2} and B = {x, y, z}
Properties of Cartesian Product
Various properties of cartesian product includes,
1. Cartesian Product is non-commutative: A × B ≠ B × A
Example:
A = {1, 2} , B = {a, b}
A × B = {(1, a), (1, b), (2, a), (2, b)}
B × A = {(a, 1), (b, 1), (b, 1), (b, 2)}
Therefore as A ≠ B we have A × B ≠ B × A
2.A × B = B × A, only if A = B
Proof:
Let A × B = B × A then we have
A ⊆ B and B ⊆ A, it follows that A = B
3. Cardinality of Cartesian Product is defined as number of elements in A × B and is equal to the product of cardinality of both sets i.e.,
|A × B| = |A| × |B|
Proof:
Let a ∈ A then the number of ordered pair (a, b) such that b ∈ B is |B|
Therefore we have |B| choices for b for each a where a ∈ A therefore the number of element in A × B is |A| × |B|
4. A × B = ∅, if either A = ∅ or B = ∅
Proof:
Suppose A×B=∅. This means there are no ordered pairs (a,b) where a∈A and b∈B.
If A is non-empty, then there exists at least one element a∈A. For any such a, there should be an ordered pair (a,b) for some b∈B, as B is not empty. But since we have assumed A×B=∅, this is a contradiction. Hence, A must be empty.
Similarly, if B is non-empty, then there exists at least one element b∈B. For any such b, there should be an ordered pair (a,b) for some a∈A, as A is not empty. But since we have assumed A×B=∅, this is a contradiction. Hence, B must be empty.
Therefore, if ? × ? = ∅, either A or B must be empty
Hence, the statement ? × ? = ∅ if and only if either A=∅ or ? = ∅ is proven.
Problems on Cartesian Product of Sets
Problem 1: Find the value of x and y given (2x – y, 25) = (15, 2x + y)?
Solution:
As we know from the property of ordered pairs, 2x – y = 15 and 25 = 2x + y.
Solving the linear equations we have x = 10 and y = 5.
Problem 2. Given A = {2, 3, 4 , 5} and B = {4 , 16 , 23}, a ∈ A, b ∈ B, find the set of ordered pairs such that a2 < b?
Solution:
As 22 < 16 and 23, 32 < 16 and 23, 42 < 23
We have the set of ordered pairs such that a2 < b is {(2, 16), (2, 23), (3, 16), (2, 23), (4, 23)}
Problem 3. If A = {9, 10} and B = {3, 4, 6}, find A × B and |A × B|?
Solution:
A × B = {(9, 3), (9, 4), (9, 6), (10, 3), (10, 4), (10, 6)}
|A × B| = |A| * |B| = 2 * 3 = 6
Problem 4. If A × B = {(a, x), (a, y ), (b, x ), (b, y)}, find A and B?
Solution:
We know A is the set of all first components in ordered pairs of A × B and
B is the set of the second component in the ordered pair of A × B.
Therefore A = {a, b} and B = {x, y}
Problem 5. Given A × B has 15 ordered pairs and A has 5 elements, find the number of elements in B?
Solution:
We know |A × B| = |A| * |B|, 15 = 5 * |B|
Therefore B has 15 / 5 = 3 elements.
Conclusion
The Cartesian Product of Sets is a fundamental concept in set theory and mathematics that helps in understanding the combination of elements from the two or more sets. By creating ordered pairs from the elements of the sets it provides a structured way to explore relationships and combinations. The practice problems presented above illustrate the various scenarios where the Cartesian Product can be applied ranging from the simple sets to more complex combinations.
FAQs on Cartesian Product of Sets
What is cartesian product of two sets i.e. A × B?
Cartesian product of sets A and B, denoted A×B, is the set of all possible ordered pairs where the first element is from A and the second from B.
Define ordered pair.
An ordered pair is a pair of elements (a, b) in which the order of the elements is significant. This means (a, b) is distinct from (b, a) if a is not equal to b.
What is cartesian product of 3 sets?
Cartesian product of three sets A, B, and C is the set of all possible ordered triples where first element is from A, second from B, and third from C.
Write formula for cartesian product of sets.
Cartesian product of two sets A and B is defined as:
A × B = {(a, b) | a ∈ A and b ∈ B}
For three sets A, B, and C:
A × B × C = {(a, b, c) | a ∈ A, b ∈ B, and c ∈ C}
What is cartesian product of a set and a null set?
Cartesian product of a set A and a null set (∅) is always an empty set (∅), as there are no elements in the null set to form pairs with elements from set A.
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Er. Neeraj K.Anand is a freelance mentor and writer who specializes in Engineering & Science subjects. Neeraj Anand received a B.Tech degree in Electronics and Communication Engineering from N.I.T Warangal & M.Tech Post Graduation from IETE, New Delhi. He has over 30 years of teaching experience and serves as the Head of Department of ANAND CLASSES. He concentrated all his energy and experiences in academics and subsequently grew up as one of the best mentors in the country for students aspiring for success in competitive examinations.
In parallel, he started a Technical Publication "ANAND TECHNICAL PUBLISHERS" in 2002 and Educational Newspaper "NATIONAL EDUCATION NEWS" in 2014 at Jalandhar. Now he is a Director of leading publication "ANAND TECHNICAL PUBLISHERS", "ANAND CLASSES" and "NATIONAL EDUCATION NEWS".
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CBSE Class 11 Maths Syllabus for 2023-24 with Marking Scheme
CBSE syllabus for class 11 Maths is divided into 5 units. The table below shows the units, number of periods and marks allocated for maths subject. The maths theory paper is of 80 marks and the internal assessment is of 20 marks.
No.
Units
Marks
I.
Sets and Functions
23
II.
Algebra
25
III.
Coordinate Geometry
12
IV.
Calculus
08
V.
Statistics and Probability
12
Total Theory
80
Internal Assessment
20
Grand Total
100
2025-26 CBSE Class 11 Maths Syllabus
Below you will find the CBSE Class Maths Syllabus for students.
Unit-I: Sets and Functions
1. Sets
Sets and their representations, empty sets, finite and infinite sets, equal sets, subsets, and subsets of a set of real numbers, especially intervals (with notations), universal set, Venn diagrams, union and intersection of sets, difference of sets, complement of a set and properties of complement.
2. Relations & Functions
Ordered pairs, Cartesian product of sets, number of elements in the Cartesian product of two finite sets, Cartesian product of the set of reals with itself (upto R x R x R), definition of relation, pictorial diagrams, domain, co-domain and range of a relation. Function as a special type of relation. 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 from one measure to another, definition of trigonometric functions with the help of unit circle, truth of the identity, signs of trigonometric functions, domain and range of trigonometric functions and their graphs, expressing sin (x±y) and cos (x±y) in terms of sinx, siny, cosx & cosy and their simple applications.
Unit-II: Algebra
1. Complex Numbers and Quadratic Equations
Need for complex numbers, especially√−1, to be motivated by the inability to solve some of the quadratic equations. Algebraic properties of complex numbers, Argand plane.
2. Linear Inequalities
Linear inequalities, algebraic solutions of linear inequalities in one variable and their representation on the number line.
3. Permutations and Combinations
The fundamental principle of counting. Factorial n. (n!) Permutations and combinations, derivation of Formulae for nPr and nCr and their connections, simple applications.
4. Binomial Theorem
Historical perspective, statement and proof of the binomial theorem for positive integral indices, Pascal’s triangle, simple applications.
5. Sequence and Series
Sequence and series, arithmetic progression (A. P.), arithmetic mean (A.M.), geometric progression (G.P.), general term of a G.P., sum of n terms of a G.P., infinite G.P. and its sum, geometric mean (G.M.), relation between A.M. and G.M.
Unit-III: Coordinate Geometry
1. Straight Lines
Brief recall of two-dimensional geometry from earlier classes. Slope of a line and angle between two lines. Various forms of equations of a line: parallel to axis, point-slope form, slope-intercept form, two-point form, intercept form and normal form. General equation of a line. Distance of a point from a line.
2. Conic Sections
Sections of a cone: circles, ellipse, parabola, hyperbola, a point, a straight line and a pair of intersecting lines as a degenerated case of a conic section. Standard equations and simple properties of parabola, ellipse and hyperbola. Standard equation of a circle.
3. Introduction to Three-Dimensional Geometry
Coordinate axes and coordinate planes in three dimensions. Coordinates of a point. Distance between two points.
Unit-IV: Calculus
1. Limits and Derivatives
Derivative introduced as rate of change both as that of distance function and geometrically, intuitive idea of limit, limits of polynomials and rational functions trigonometric, exponential and logarithmic functions, definition of derivative relate it to the slope of the tangent of the curve, derivative of sum, difference, product and quotient of functions. Derivatives of polynomial and trigonometric functions.
Unit-V: Statistics and Probability
1. Statistics
Measures of Dispersion: Range, mean deviation, variance and standard deviation of ungrouped/grouped data.
2. Probability
Events; occurrence of events, ‘not’, ‘and’ and ‘or’ events, exhaustive events, mutually exclusive events, Axiomatic (set theoretic) probability, connections with other theories of earlier classes. Probability of an event, probability of ‘not’, ‘and’ and ‘or’ events.
Students can also get the syllabus of all the subjects by visiting CBSE Class 11 Syllabus page. Learn Maths & Science in an interactive & fun-loving way with Anand Classes App/Tablet.
Frequently Asked Questions on CBSE Class 11 Maths Syllabus 2025-26
Q1
What is the marks distribution for internals and theory exams according to the CBSE Maths Syllabus for Class 11?
The marks distribution for internals is 20 marks and the theory exam is 80 marks based on the CBSE Class 11 Maths Syllabus.
Q2
Which is the most important chapter in the CBSE Class 11 Maths Syllabus?
The important chapter in the CBSE Class 11 Maths Syllabus is Algebra which is for 25 marks in the overall weightage.
Q3
What are the chapters covered in Unit III of the CBSE Class 11 Maths Syllabus?
The chapters covered in Unit III of the CBSE Class 11 Maths Syllabus are straight lines, conic sections and an introduction to three-dimensional geometry.
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