Relations on cartesian product

Question

The relation $T$ is defined as follows: $T \subseteq \mathbb{R}\times \mathbb{R} : xTy \Leftrightarrow y > x + 1$.

Is $T$ a function and why?

Thanks.

Answer 1

I assume that by $R$, you mean the real numbers. For $T$ to be a function, you need the following:

1) All $x$ in $\mathbb{R}$ are related to a $y$ in $\mathbb{R}$.

2) No $x$ in $\mathbb{R}$ is related to two different $y_1,y_2$ in $\mathbb{R}$.

Let me get you started. No matter what $x$ you choose, will there be a $y$, such that $y>x+1$? This is property 1. If you take $x\in\mathbb{R}$, will there exist two different $y_1, y_2$, such that both $y_1>x+1$ and $y_2>x+1$ are true? If such $y$'s exist, then it violates property 2.