Question

Let R be the relation on the set A = {1, 2, 3, 4, 5, 6, 7} defined by the rule

if the integer (a – b) is divisible by 4. List the elements of R and its

inverse?

Answer 1

Let $R$ be the relation on the set $A =\{1, 2, 3, 4, 5, 6, 7\}$ defined by the rule if the integer $(a -b)$ is divisible by 4. Let us list the elements of $R$:

$R=\{(1,5),(5,1),(2,6),(6,2),(3,7),(7,3)\}$.

It follows from $(a,b)\in R$ that $(a -b)$ is divisible by 4, and hence $(b -a)=-(a-b)$ is also divisible by 4. Consequently, $(b,a)\in R.$ We conclude that the for the inverse relation we have $R^{-1}=\{(a,b)\ | \ (b,a)\in R\}=R.$