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 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.”
Definition: “A relation R on a set A is called transitive if whenever (a,b)∈R and (b,c)∈R, then (a,c) ∈ R, for all a,b,c ∈ A.”
The definitions above are from the textbook “Discrete Mathematics and its applications” By Rosen.
In this case, there are many possible answers. I’ll just show you two and you can find more of them.
Buildings that have the same color
Reflexive: Every building has the same color of itself. Therefore, it is reflexive.
Symmetric: If building a has the same color as building b, then building b has also the same color as building a. Therefore, it is symmetric.
Transitive: If building a has the same color as building b, and building b has the same color as building c, then building a has the same color as building c. Therefore, it is transitive.
This relation is an equivalence relation because it is reflexive, symmetric, and transitive.
An equivalence class consists of the set of buildings that have the same color.
Buildings that have the same number of floors (or stories)
Reflexive: Every building has the same number of floors as itself. Therefore, it is reflexive.
Symmetric: If building a has the same number of floors as building b, then building b has also the same number of floors as building a. Therefore, it is symmetric.
Transitive: If building a has the same number of floors as building b, and building b has the same number of floors as building c, then building a has the same number of floors as building c. Therefore, it is transitive.
This relation is an equivalence relation because it is reflexive, symmetric, and transitive.
An equivalence class in this case will consist of the set of buildings that have the same number of floors.
Related posts:
- 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)
- 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?
- 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
- Define three equivalence relations on the set of classes offered at your school. Determine the equivalence classes for each of these equivalence relations
- 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
