Question

A car dealership sells cars that were made in 2015 through 2020. Let the cars for sale be the domain of a relation R where two cars are related if they were made in the same year. (a) Prove that this relation is an equivalence relation. (b) Describe the partition defined by the equivalence classes.

Answer 1

The given relation obeys reflexive, symmetric, and transitive properties, therefore, it is in relation to equivalence relation.

How is the equivalence relation depicted?

Let the domain of the relation is car(x) belongs to 2015 , 2016, 2017, 2018, 2019, 2020

To prove equivalence relation, we need to prove reflexive, symmetric, transitive properties.

Reflexive relation: x is related to x

Here car(x) is made in 2015, so, car(x)) is in R where R is reflexive relation.

Symmetric relation: x is related to y implies y is related to x

Here (car(x) , car(y)) is in R , i.e car(x) and car(y) are made in same year. R obeys symmetric property

Transitive property: x is related to y and y is related to z then x is related to z. Here (car(x),car(y)), (car(y), car(z)) is in R. Therefore (car(x) , car(z)) is in R

Therefore, the given relation obeys reflexive, symmetric, and transitive properties, therefore, it is in relation to equivalence relation.

B. The cars which are made in the same year are in one equivalence class all cars are made in overall 6 years. We get 6 equivalence classes. These classes are partitions, so we have six partitions

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