Relation Problem?

Question

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

R = {(a,b) (element symbol)AxA׀ a + b <= 9}

(a) List the elements of R.

My Answer:

R = {(1,1),(1,2),(1,3),
(1,4),(1,5),(1,6),
(2,1)(2,2),(2,3),
(2,4),(2,5),(2,6),
(3,1),(3,2),(3,3),
(3,4),(3,5),2,6),
(4,1),(4,2),(4,3),
(4,4),(4,5),(5,1),
(5,2),(5,3),(5,4),
(6,1),(6,2),(6,3)}

(b) Let Δ = {(x,x) ׀ x (element symbol)A } Is Δ (subset symbol) of R?

My Answer so far:

Δ = {(1,1),(2,2),(3,3)
,(4,4),(5,5)}

Not sure how to explain it?

(c) R = R^(-1)? Not sure how to show it?

Thank you for your help!

Answer 1

(a) is OK.

(b) Δ = {(1,1),(2,2),(3,3), ,(4,4),(5,5), (6,6)} - this is not a subset of R because elementt (5,5) and (6,6) of Δ are not elements of R.

(c) This follows from the commutativity law:
(a,b) E R ↔ a+b=b+a<=9 ↔ (b,a) E R ↔ (a,b) E R^-1