Suppose that R is the following relation on the set A1234 RA

Suppose that R is the following relation on the set A={1,2,3,4}: R=A×A{(1,1),(2,2),(3,3),(4,4)}

Determine which of the following is the relation R2.

R1=A×A

R2={}

R3=R

Answer:

Let A={1,2,3,4} then AxA={(1,1),(1,2),(1,3),(1,4),(2,1),(2,2),(2,3),(2,4),(3,1),(3,2),(3,3),(3,4),(4,1),(4,2),(4,3),(4,4)}

let R be the relation defined on the set A={1,2,3,4} such that R=AxA - {(1,1),(2,2),(3,3),(4,4)}

That is R={(1,2),(1,3),(1,4),(2,1),(2,3),(2,4),(3,1),(3,2),(3,4),(4,1),(4,2),(4,3)}

Since any subset of AxA is a relation from A to A. So that R1=AxA is a relation ,since every set is subset to itself.

R2={ } is relation from A to A since empty set is subset to every set.

R3=R is a subset of AxA hence R3 is a relation from A to A