HOLIDAYS HOMEWORK MATHS

Chapter 1
1mark
1. If and the composite function then is equal to_________________.

2. Let R be the relation in the set Z of integers given by R= {(): 3 divides}.
Show that the relation R symmetric? Write the equivalence class [1].

3. Examine whether the operation * defined on R by a * b = ab + 1 is (i) a
binary or not. (ii) if a binary operation, is it associative or not ?
(2019BVM/65/1/1)

4. Examine whether the operation * defined on , the set of all real numbers, by
= is a binary operation or not, and if it is a binary operation, find whether it
is associative or not. (2019BVM/65/2/1)

5. If * is defined on the set R of all real numbers by *: a * b = , find the
identity element, if it exists in with respect to *. (2019BVM/65/3/1.3)
6. Let * be a binary operation on R – {– 1} defined by , for all a, b R – {–
1}.Show that * is neither commutative nor associative in R – {– 1}.
(2019/BVM/65/3/2)

4 marks

7. Let f: A → B be a function defined as f(x) =, where and B = . Is the function
f one - one and onto? Is f invertible? If yes, then find its inverse.
4

8. Show that the relation R on defined as R = {(a, b): a ≤ b}, is reflexive, and
transitive but not symmetric. (2019BVM/65/1/1)
OR
Prove that the function f: N N, defined by f(x) = x 2 + x + 1 is one-one but
not onto. Find inverse of f: N S, where S is range of f. (2019BVM/65/1/1)

9. Check whether the relation R defined on the set A = {1, 2, 3, 4, 5, 6} as
R={(a,b) : b = a + 1} is reflexive, symmetric or transitive.
(2019BVM/65/2/1)
OR

Let f: N Y be a function defined as f(x) = 4x + 3, where Y = {y N: y = 4x+
3, for some x N}. Show that f is invertible. Find its inverse.
(2019BVM/65/2/1)

10. Show that the relation R on the set Z of all integers, given by
R = {(a, b) : 2 divides (a – b)} is an equivalence relation.
(2019BVM/65/3/1.2.3)
OR
If f(x) = , show that fof(x) = x for all . Also, find the inverse of f.
(2019BVM/65/3/1.2.3)