Let R be the real line. Let the relations S and T or R be defined by

$$S = \{ (x,y):y = x + 1,0 < x < 2\} ,T = \{ (x,y):x - y$$ is an integer}. Then

Let the relation p be defined on R by a p b holds if and only if a $$ - $$ b is zero or irrational, then

Let p₁ and p₂ be two equivalence relations defined on a non-void set S. Then

Let the relation $$\rho $$ be defined on R as a$$\rho $$b if 1 + ab > 0. Then,