Relation and Function
NCERT Exemplar Problem
Long Answer Type Part 3
Question 22: Each of the following defines a relation on N:
Determine which of the above relations are reflexive, symmetric and transitive.
Solution:
Hence, the given relation is only transitive.
Therefore, R is not reflexive.
Therefore, R is symmetric.
Therefore, R is not transitive.
Thus, R is only symmetric.
Therefore, R is reflexive.
Therefore, it is clear that R is symmetric.
Therefore, R is transitive.
Therefore, R is not reflexive.
Therefore, R is not symmetric.
Since, there is no element which begins with
Therefore, R is a transitive.
Question 23: Let A = {1, 2, 3, ………, 9} and R be the relation in A x A defined by (a, b) R (c, b) if a + d = b + c for (a, b), (c, d) in A × A. Prove that R is an equivalence relation and also obtain the equivalent class [(2, 5)].
Solution:
Therefore, R is reflexive.
Let (a, b) R (c, d)
Therefore, R is symmetric.
Let (a, b) R (c, d) and (c, d) R (e, f)
Therefore, R is transitive.
Thus, R is reflexive, symmetric and transitive.
Therefore, R is an equivalence relation.
Equivalence class containing {(2, 5)} is {(1, 4), (2, 5), (3, 6), (4, 7), (5, 8), (6, 9)}.
Question 24: Using the definition, prove that the function
is invertible if and only if f is both one-one and onto.
Solution: By the definition of an invertible function:
A function
is defined to be and invertible function, if there exists a function
The function g is called the inverse of f and is denoted by f – 1.
has to one-one and onto.
Therefore, f(x) should be both one-one and onto.