In each of 3-6, the relation R is an equivalence relation on A. As in Example 8.3.5, first find the specified equaivalence classes. Then state the number of distinct equivalenvce classes for R and list them.

A = { 1 , 2 , 3 , 4 , ... , 20 } . R is defined on A as folloes:

For   all   x , y ∈ A ,         x R y     ⇔ 4 | ( x − y )

Equivalence classes:[1], [2], [3], [4], [5].

To find the distinct equivalence classes of R.

Given information:

The relation R is an equivalence relation on the set A. A={1,2,3,4,...,20}. R is defined on A as follows:                 For all x,y∈A, x R y ⇔ 4|(x−y).

Calculation:

A={1,2,3,4,…,20}R={(x,y)∈A×A|4( x−y)}

Let us first group the ordered pairs in R that have at least one common element (with at least one of the other elements in the group).

4|(x−y) is true if and only if x−y is a multiple of 4 if and only if x and y differ by 0, 4, 8, 12 or 16 (as the elements are between 1 and 20).

Thus (x−y)∈R when x and y differ by 0, 4, 8, 12 or 16. First group: (1,1),(1,5),(1,9),(1,13),(1,17)                   (5,1),(5,5),(5,9),(5,13),(5,17)                   (9,1),(9,5),(9,9),(9,13),(9,17)                   (13,1),(13,5),(13,9),(13,13),(13,17)                   (17,1),(17,5),(17,9),(17,13),(17,17)

Second group: (2,2),(2,6),(2,10),(2,14),(2,18)                        (6,2),(6,6),(6,10),(6,14),(6,18)                        (10,2),(10,6),(10,10),(10,14),(10,18)                        (14,2),(14,6),(14,10),(14,14),(14,18)                        (18,2),(18,6)(