Prove that R is an equivalence relation. (b) Enumerate all possible equivalence classes in R. (As per lecture, any equivalence class is the set of all elements in W that are related to each other via R.) g

Income Question

Given that

A = {0,1,2,3,4}

R = { (0,0), (0,4), (1,1), (1,3), (2,2), (3,1), (3,3), (4,0), (4,4) }

Answer:

See Explanation Below

Step-by-step explanation:

a.

The equivalence class of A, is the set of all elements x in A such that x is related to A by R

Let ~ be a equivalence relation on set A such that x ∈ A and y ∈ A such that x ~ y.

Meaning that x is an element of A and y is an element of A where x = y

Considering the given data.

A = {0,1,2,3,4}

R = { (0,0), (0,4), (1,1), (1,3), (2,2), (3,1), (3,3), (4,0), (4,4) }.

We find that;

(0,4) ∈ R; so 0 and 4 belong to a class

(1,3) ∈ R; so 1 and 3 belong to a class

(2,2) ∈ R; 2 doesn't occur in any other pair in R; So, 2 is in its own class.

To prove that R is an equivalence relation;.

We set A/R

A/R gives { (0,4), (1,3), (2,2) }.

So, R is an equivalence of A.

b. All possible equivalence classes in R = { (0,4), (1,3), (2,2) }