I have only written proofs that prove relations using equality are transitive. I have no idea how to manipulate equations with inequalities.
R = {(x, y) | x − y > 1} is a relation on ℝ
Claim: R is transitive
Proof:
Choose x, y, z ∈ ℝ and assume R(x,y) and R(y,z)
So x - y > 1 and y - z > 1
Stuck..
I know I need to prove x - z > 1 but I dont know how to get there.
I figured it out thanks to the replies. I just had to add the two inequalities.
Choose x, y, z ∈ ℝ and assume R(x,y) and R(y,z)
So x - y > 1 and y - z > 1
(x - y) + (y - z) > 2
x - z > 2 > 1
Therefore R(x,z)