Types of Relations in Set Theory

Questions and Answers

Explain why the relation R = {(x, y) : 3x − y = 0} is not reflexive on the set A = {1, 2, 3 … 13, 14}.

The relation R is not reflexive because for a relation to be reflexive, it must contain all pairs (a, a) where a is an element of the set. In this case, R does not contain pairs like (1, 1), (2, 2) ... (14, 14), so it is not reflexive.

Why is the relation R = {(x, y) : 3x − y = 0} not symmetric?

The relation R is not symmetric because although (1, 3) is in R, (3, 1) is not in R. For a relation to be symmetric, if (a, b) is in R, then (b, a) must also be in R, which is not the case here.

Explain why the relation R = {(x, y) : 3x − y = 0} is not transitive.

The relation R is not transitive because although (1, 3) and (3, 9) are in R, (1, 9) is not in R. For a relation to be transitive, if (a, b) and (b, c) are in R, then (a, c) must also be in R, which is not the case here.

Determine whether the relation R = {(x, y) : y = x + 5 and x < 4} on the set of natural numbers is reflexive, symmetric, and transitive.

The relation R is not reflexive because it does not contain pairs like (1, 1), (2, 2), (3, 3), etc. It is also not symmetric because it contains pairs like (1, 6), but not (6, 1). Additionally, it is not transitive because, for example, (1, 6) and (6, 11) are in R, but (1, 11) is not in R. Signup and view all the answers

State whether the relation R = {(x, y) : y = x + 5 and x < 4} on the set of natural numbers is reflexive, symmetric, or transitive, and provide a brief explanation.

The relation R is neither reflexive, symmetric, nor transitive. It is not reflexive because it does not contain pairs like (1, 1), (2, 2), (3, 3), etc. It is not symmetric because it contains pairs like (1, 6), but not (6, 1). Additionally, it is not transitive because, for example, (1, 6) and (6, 11) are in R, but (1, 11) is not in R. Signup and view all the answers

Match the following properties with their definitions in relation to sets:

Reflexive = Every element is related to itself Symmetric = If (a, b) is in the relation, then (b, a) must also be in the relation Transitive = If (a, b) and (b, c) are in the relation, then (a, c) must also be in the relation Inverse of a relation = A new relation obtained by swapping the elements of each pair in the original relation Signup and view all the answers

Match the following terms with their definitions related to relations and sets:

Directed graph = A graph where edges have a direction associated with them Set A = {1, 2, 3 … 13, 14} = A set containing elements from 1 to 14 Natural numbers = Positive integers including zero Relation R = {(x, y): y = x + 5 and x < 4} = A relation defined on a set of elements where y is x+5 and x is less than 4 Signup and view all the answers

Match the following examples with their respective properties in relation to sets:

Example -6- = Inverse of a relation Example -7- = Directed graph on a relation Example -9- = Properties of a relation Example -11- = Neither reflexive, nor symmetric, nor transitive Signup and view all the answers

Match the following terms with their relevant applications in programming:

General-purpose programming = Python Client-side scripting for web applications = JavaScript Database queries = SQL Styling web pages = CSS Signup and view all the answers

Match the following relations with their corresponding properties:

Relation R = {(x, y): 3x − y = 0} = Not reflexive, not symmetric, not transitive Relation R = {(x, y): y = x + 5 and x < 4} = Reflexive: No; Symmetric: No; Transitive: No Relation R in the set A = {1, 2, 3…13, 14} defined as R = {(x, y): 3x − y = 0} = Reflexive: No; Symmetric: No; Transitive: No Relation R in the set N of natural numbers defined as R = {(x, y): y = x + 5 and x < 4} = Reflexive: No; Symmetric: No; Transitive: No Signup and view all the answers

Which of the following best describes the relation R = {(x, y) : 3x − y = 0} on the set A = {1, 2, 3 … 13, 14}?

Not reflexive, symmetric, or transitive Signup and view all the answers

In the relation R = {(x, y) : y = x + 5 and x < 4} on the set of natural numbers, which property is violated?

Transitivity Signup and view all the answers

Why is the relation R = {(x, y) : y = x + 5 and x < 4} not reflexive?

It does not include (1, 1) Signup and view all the answers

In the set A = {1, 2, 3 … 13, 14}, why is the relation R = {(x, y) : 3x − y = 0} not transitive?

(1, 3), (3, 9) ∈R, but (1, 9) ∉ R Signup and view all the answers

Which property is violated by the relation R = {(x, y) : y = x + 5 and x < 4} on the set of natural numbers?

Symmetry Signup and view all the answers