Question

Let R be the relation on the set A = {1, 2, 3, 4, 5, 6, 7} defined by  the rule (a b R, ) if the integer product of (ab)        is divisible by 4. List the elements of R and its inverse?  

Answer 1

1*1=1 is not divisible by 4, therefore $(1,1) \notin R$

1*2=2*1=2 is not divisible by 4, therefore $(1,2),\,(2,1) \notin R$

1*3=3*1=3 is not divisible by 4, therefore $(1,3),\,(3,1) \notin R$

1*4=4*1=4 is divisible by 4, therefore $(1,4),\,(4,1) \in R$

1*5=5*1=5 is not divisible by 4, therefore $(1,5),\,(5,1) \notin R$

1*6=6*1=6 is not divisible by 4, therefore $(1,6),\,(6,1) \notin R$

1*7=7*1=7 is not divisible by 4, therefore $(1,7),\,(7,1) \notin R$

Continuing the reasoning in a similar way, we get the elements of the relation R:

$R = \{ (1,4),\,(4,1),\,(2,2),\,(2,4),\,(4,2),\,(2,6),\,(6,2),\,(3,4),\,(4,3),\,(4,4),$

$(4,5),\,(4,6),\,(4,7),\,(5,4),\,(6,4),\,(7,4),\,(6,6)\}$

Find the inverse relation:

${R^{ - 1}} = \{ (y,x)|(x,y) \in R\} = \{ (4,1),\,(1,4),\,(2,2),\,(4,2),\,(2,4),\,(6,2),\,(2,6),\,(4,3),\,(3,4),\,(4,4),\,(5,4),\,$

$(6,4),\,(7,4),\,(4,5),\,(4,6),\,(4,7),\,(6,6)\}$