Relations and Functions quiz

The relation R = {(1, 1), (2, 2), (3, 3)} on {1, 2, 3} is:

The number of relations on a set with 2 elements is:

The function f: R -> R, f(x) = x^3 is:

The modulus function f: R -> R, f(x) = abs(x) is:

The relation R = {(1, 2), (2, 1)} on {1, 2, 3} is:

The number of one-one functions from {1, 2, 3} to itself is:

The function f: R -> R, f(x) = x^2 is:

The number of reflexive relations on a set with 3 elements is:

The equivalence classes of an equivalence relation on a set are:

The relation is parallel to on the set of lines in a plane is:

Show that the relation R on R defined by R = {(x, y): x <= y} is reflexive and transitive but not symmetric. (2 marks)

Is f: N -> N, f(x) = 2x onto? Justify. (2 marks)

Write the equivalence classes of the relation x - y is even on {1, 2, 3, 4, 5}. (2 marks)

Give an example of a function that is onto but not one-one. (2 marks)

Show that the relation is perpendicular to on the set of lines in a plane is symmetric but not transitive. (2 marks)

If A has m elements and B has n elements, how many relations are there from A to B? (2 marks)

Show that f: R -> R, f(x) = 5x + 2 is one-one. (2 marks)

Check whether R = {(x, y): x divides y} on N is symmetric. (2 marks)

Can a function from a set of 4 elements to a set of 3 elements be one-one? Explain. (2 marks)

Define the universal relation and show that it is an equivalence relation. (2 marks)

Check whether the relation R = {(1, 1), (2, 2), (3, 3), (1, 2), (2, 3)} on {1, 2, 3} is reflexive, symmetric or transitive. (3 marks)

Show that f: R0 -> R0 defined by f(x) = 1/x, where R0 is the set of non-zero real numbers, is one-one and onto. (3 marks)

Show that f: R -> R, f(x) = 2x + 3 is a bijection and find the element whose image is 11. (3 marks)

A has 4 elements and B has 3 elements. Find the number of functions from A to B, the number of one-one functions and the number of onto functions. (3 marks)

On A = {1, 2, 3, 4, 5, 6, 7}, R = {(x, y): x - y is divisible by 3}. Show that R is an equivalence relation and write all its equivalence classes. (3 marks)

Read the passage and answer the questions. In a school, a relation R is defined on the set of students by x R y if x and y have the same blood group. (i) Is R reflexive and symmetric? (ii) Is R transitive? (iii) How many equivalence classes are there if students have the blood groups A, B, AB and O only? (5 marks)

Read the passage and answer the questions. Each student of a class of 40 is assigned a unique roll number from 1 to 40. The function f maps each student to the roll number. (i) Is f one-one? (ii) Is f onto the set {1, 2, ..., 40}? (iii) What would happen if the codomain were {1, 2, ..., 50}? (5 marks)

Read the passage and answer the questions. Five cities are mapped to the states they belong to: two cities are in Kerala, two in Punjab and one in Bihar, with the codomain {Kerala, Punjab, Bihar}. (i) Is the function one-one? (ii) Is it onto? (iii) Draw the arrow diagram. (5 marks)

Read the passage and answer the questions. In a family, a relation R is defined on the set of family members by x R y if x is the brother of y. (i) Is R reflexive? (ii) Is R symmetric? (iii) Is R transitive? Give reasons. (5 marks)

Read the passage and answer the questions. A teacher defines f: Z -> Z by f(x) = x + 5 and g: Z -> Z by g(x) = 2x. (i) Is f a bijection? (ii) Is g onto? (iii) Which of these functions is one-one? (5 marks)

Prove that the relation R on Z defined by R = {(x, y): x - y is divisible by 5} is an equivalence relation, and write its equivalence classes. (6 marks)

Show that f: N -> N, f(x) = 2x is one-one but not onto, and that g: N -> N, g(1) = 1 and g(n) = n - 1 for n >= 2, is onto but not one-one. (6 marks)

Show that f: R -> R, f(x) = x^2 is neither one-one nor onto, and that f: [0, infinity) -> [0, infinity), f(x) = x^2 is a bijection. Draw graphs. (6 marks)

Let R be the relation on N x N defined by (p, q) R (r, s) if p + s = q + r. Prove that R is an equivalence relation. (6 marks)

Let A = R - {3} and B = R - {1}. Show that f: A -> B defined by f(x) = (x - 2)/(x - 3) is one-one and onto. (6 marks)