# Relations

## Give an example of a relation on a set that is a) both symmetric and antisymmetric. b) neither symmetric nor antisymmetric

Relevant definitions: Definition: “A relation R on a set A is called symmetric if (b, a) ∈ R whenever (a,b) ∈ R, for all a,b ∈ A. A relation R on a set A such that for all a, b ∈ A, if (a, b) ∈ R and (b, a) ∈ R, then a = […]

## Show that the relation R = ∅ on a nonempty set S is symmetric and transitive, but not reflexive

Let’s refresh the definitions that are relevant to this exercise. Definition: “A relation R on a set A is called reflexive if (a, a) ∈ R for every element a ∈ A.” Definition: “A relation R on a set A is called symmetric if (b, a) ∈ R whenever (a,b) ∈ R, for all a,b

## Determine whether the relation R on the set of all real numbers is reflexive, symmetric, antisymmetric, and/or transitive, where (x,y)∈R if and only if

Let’s start with relevant definitions. Definition: “A relation R on a set A is called reflexive if (a, a) ∈ R for every element a ∈ A.” Definition: “A relation R on a set A is called symmetric if (b, a) ∈ R whenever (a,b) ∈ R, for all a,b ∈ A. A relation R

## a) List all the ordered pairs in the relation R = {(a,b) | a divides b} on the set {1,2,3,4,5,6}.  b) Display this relation graphically, as was done in Example 4. c)Display this relation in tabular form, as was done in Example 4

To list the ordered pairs, we must follow the condition stated in the relation. In this case, (a,b) ∊R if a divides b. In other words, when we divide b by a, we get as a result an integer value, and the remainder is 0. a) List all the ordered pairs in the relation R

## Show that the relation R=∅ on the empty set S=∅ is reflexive, symmetric, and transitive

Let’s refresh the relevant definitions that will help us to solve this exercise. Definition: “A relation R on a set A is called reflexive if (a, a) ∈ R for every element a ∈ A.” Definition: “A relation R on a set A is called symmetric if (b, a) ∈ R whenever (a,b) ∈ R,

## Determine whether the relation R on the set of all Web pages is reflexive, symmetric, antisymmetric, and/or transitive, where (a, b) ∈ R if and only if

Let’s start with the relevant definitions that will help us to solve this exercise. Definition: “A relation R on a set A is called reflexive if (a,a) ∈ R for every element a ∈ A.” Definition: “A relation R on a set A is called symmetric if (b,a) ∈ R whenever (a,b) ∈ R, for

## Suppose that A is a nonempty set and R is an equivalence relation on A. Show that there is a function f with A as its domain such that (x,y) ∈ R if and only if f(x) = f(y)

As usual, let’s start with the relevant definitions. Definition: “A relation on a set A is called an equivalence relation if it is reflexive, symmetric, and transitive.” Definition: “A relation R on a set A is called reflexive if (a, a) ∈ R for every element a ∈ A.” Definition: “A relation R on a

## Let R be the relation on the set of all sets of real numbers such that SRT if and only if S and T have the same cardinality. Show that R is an equivalence relation. What are the equivalence classes of the sets {0, 1, 2} and Z?

See below the relevant definitions that we need to know to solve this exercise. Definition: “A relation on a set A is called an equivalence relation if it is reflexive, symmetric, and transitive.” Definition: “A relation R on a set A is called reflexive if (a, a) ∈ R for every element a ∈ A.”

## Define three equivalence relations on the set of students in your discrete mathematics class different from the relations discussed in the text. Determine the equivalence classes for each of these equivalence relations

See below the relevant definitions that we need to know to solve this exercise. Definition: “A relation on a set A is called an equivalence relation if it is reflexive, symmetric, and transitive.” Definition: “A relation R on a set A is called reflexive if (a,a) ∈ R for every element a ∈ A.” Definition:

## Define three equivalence relations on the set of classes offered at your school. Determine the equivalence classes for each of these equivalence relations

As usual, let’s start with the relevant definitions. Definition: “A relation on a set A is called an equivalence relation if it is reflexive, symmetric, and transitive.” Definition: “A relation R on a set A is called reflexive if (a,a) ∈ R for every element a ∈ A.” Definition: “A relation R on a set