Relations and Functions

Topics in this chapter

• Cartesian Products and Relations
• Relation Properties and Classification
• Equivalence Relations and Classes
• Partial Orders and Divisibility
• Counting Relations
• Functions and Their Types
• Composition and Inverse Functions

Cheat sheet

Practice questions

1. Question 1Q-8DC1C2

   Let f: R → R be defined by f(x) = x². Which of the following describes f?

   1. A.It is both one-one and onto.
   2. B.It is one-one but not onto.
   3. C.It is neither one-one nor onto.
   4. D.It is onto but not one-one.
   See the answer and solution
2. Question 2Q-041D33

   On the set of integers Z, define a relation R by aRb if and only if 3 divides (a − b). Which statement is correct?

   1. A.R is reflexive but neither symmetric nor transitive.
   2. B.R is symmetric and transitive but not reflexive.
   3. C.R is an equivalence relation.
   4. D.R is reflexive and symmetric but not transitive.
   See the answer and solution
3. Question 3Q-276BC0

   Let f: R \ {−2} → R \ {1} be defined by f(x) = (x − 1)/(x + 2). If f is invertible, which is the correct expression for f⁻¹?

   1. A.f⁻¹(x) = (1 − 2x)/(1 + x)
   2. B.f⁻¹(x) = (1 + 2x)/(1 − x)
   3. C.f⁻¹(x) = (x + 1)/(2 − x)
   4. D.f⁻¹(x) = (2x − 1)/(x + 1)
   See the answer and solution
4. Question 4Q-85750D

   On Z, define a relation R by aRb if and only if a − b is even. Prove that R is an equivalence relation and find its equivalence classes.

   See the answer and solution
5. Question 5Q-A11B12

   Let A = {1, 2, 3}, B = {a, b}, and R = {(1, a), (2, b), (3, a)} be a relation from A to B. Which statement is correct?

   1. A.R is not a function from A to B.
   2. B.R is a function, one-one and onto.
   3. C.R is a function, onto but not one-one.
   4. D.R is a function, one-one but not onto.
   See the answer and solution