Discrete Mathematics

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

Relations /

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, […]

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

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

Relations /

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

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 Read More »

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)

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

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) Read More »

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?

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.”

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? Read More »

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

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 students in your discrete mathematics class different from the relations discussed in the text. Determine the equivalence classes for each of these equivalence relations Read More »

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

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

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

Define three equivalence relations on the set of buildings on a college campus. Determine the equivalence classes for each of these equivalence relations. 

Relations /

First, let’s take a look at 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

Define three equivalence relations on the set of buildings on a college campus. Determine the equivalence classes for each of these equivalence relations.  Read More »

Which of these relations on the set of all functions from Z to Z are equivalence relations? Determine the properties of an equivalence relation that the others lack

Relations /

Below are 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:

Which of these relations on the set of all functions from Z to Z are equivalence relations? Determine the properties of an equivalence relation that the others lack Read More »

Which of these relations on the set of all people are equivalence relations? Determine the properties of an equivalence relation that the others lack

Relations /

Let’s start with the relevant definitions that will help us 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:

Which of these relations on the set of all people are equivalence relations? Determine the properties of an equivalence relation that the others lack Read More »

Let R be the relation on the set of ordered pairs of positive integers such that ((a, b), (c, d)) ∈ R if and only if a + d = b + c. Show that R is an equivalence relation

Relations /

As usual for us, we will start refreshing the relevant definitions 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.”

Let R be the relation on the set of ordered pairs of positive integers such that ((a, b), (c, d)) ∈ R if and only if a + d = b + c. Show that R is an equivalence relation Read More »