CBSE Class 11 Mathematics Relations And Functions Notes

Class XI

Mathematics

Chapter:2 Relations and Functions

Points to Remember

Key Concepts

1. A pair of elements grouped together in a particular order is known as an ordered pair.

2. The two ordered pairs (a, b) and (c, d) are said to be equal if and only if a = c and b = d.

3. Let A and B be any two non empty sets. The Cartesian product A × B is the set of all ordered pairs of elements of sets from A and B defined as follows: A × B = {(a, b) : a ÎA, b ÎB}. Cartesian product of two sets is also known as Product Set.

4. If any of the sets of A or B or both are empty then the set A × B will also be empty and consequently, n(A × B) =0

5. If the number of elements in A is m and the number of elements in set B is n then the set A × B will have mn elements

6. If any of the sets A or B is infinite, then A × B is also an infinite set.

7. Cartesian product of sets can be extended to three or more sets If A, B and C are three non empty sets, then A × B × C = {(a, b, c): a ÎA, bÎB, cÎC}.Here (a, b, c) is known as an ordered triplet.

8. Cartesian product of a non empty set A with an empty set is empty set i.e A X Φ = Φ

9. The Cartesian product is not commutative, namely A x B is not the same as B x A, unless A and B are equal.

10.Cartesian product is associative, namely A x (B x C)=(A x B) x C

11. R × R = {(a, b) : a ÎR, b ÎR} represents the coordinates of all points in two dimensional plane. R × R × R = {(a, b, c): a ÎR, b ÎR, c ÎC} represents the coordinates of all points in three dimensional plane.

12. A relation R from the non empty set A to another non empty set B is a subset of their Cartesian product A × B, i.e R Í A × B.

13. If (x, y) ÎR or x R y then x is related to y and (x, y) ÏR or x R y then x is not related to y.

14.The second element b in the ordered pair (a,b) is the image of first element a and a is the pre-image of b.

15.The Domain of R is the set of all first elements of the ordered pairs in a relation R. In other words domain is the set of all the inputs of the relation.

16. If the relation R is from a non empty set A to non empty set B then set B is called the co - domain of relation R.

17.The set of all the images or the second element in the ordered pair (a,b) of relation R is called the Range of R.

18.The total number of relations that can be defined from a set A to a set B is the number is possible subsets of A X B.

19.A × B can have 2mn subsets. This means there are 2mn relations from A to B

20.Relation can be represented algebraically and graphically. The various methods are as follows:

21. A relation f from a non –empty set A to another non- empty set B is said to be a function if every element of A has a unique image in B.

22. The domain of f is the set A. No two distinct ordered pairs in f have the same first element.

23. Every function is a relation but converse is not true

24. If f is a function from A to B and (a, b) ∈ f, then f (a) = b, where b is called image of a under f and a is called the pre-image of b under f

25. If f: A → B A is the domain and B is the co domain of f.

26. The Range of the function is the set of images.

27. A real function has the set of real numbers or one of its subsets both as its domain and as its range.

28.Identity function: f: X → X is an identity function if f(x) = x for each
x ∈ A

29.Graph of the identity function is a straight line that makes an angle of 45o with both x and y axes. All points on this line have their x and y coordinates equal.

30.Constant function: A constant function is one that maps each element of the domain to a constant. Domain of this function is R and range is the singleton set {c} where c is a constant.

31.. Graph of constant function is a line parallel to the x axis. The graph lies above x axis if the constant c > 0, below the x axis if the constant c < 0 and is same as x axis if c = 0
 

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