Prove if the relation R is transitive then rR the reflexive

Prove if the relation R is transitive then r(R), the reflexive closure, is transitive.

Do not use specific examples in the proof.

Solution

Ans-

• Definition:
Let R be a binary relation on A.
• R is reflexive if for all x A, (x,x) R.
(Equivalently, for all x e A, x R x.)
• R is symmetric if for all x,y A, (x,y) R
implies (y,x) R. (Equivalently, for all x,y A,
x R y implies that y R x.)
• R is transitive if for all x,y,z A, (x,y) R and
(y,z) R implies (x,z) R. (Equivalently, for all
x,y,z A, x R y and y R z implies x R z.)

• Reflexive: The relation R on {1,2,3} given by
R = {(1,1), (2,2), (2,3), (3,3)} is reflexive. (All
loops are present.)
• Symmetric: The relation R on {1,2,3} given by
R = {(1,1), (1,2), (2,1), (1,3), (3,1)} is symmetric.
(All paths are 2-way.)
• Transitive: The relation R on {1,2,3} given by
R = {(1,1), (1,2), (2,1), (2,2), (2,3), (1,3)} is
transitive. (If I can get from one point to another
in 2 steps, then I can get there in 1 step.)