Let R and S be binary relations on N (repeat,N) defined as follows

Question

Let $R$ and $S$ be binary relations on $\mathbb{N}$ defined as follows:

$x \mathrel{R} y$ if $x > y+2$ and $x \mathrel{S} y$ if $x = y -2$ Let $T = R \circ S$ (composition).

1. (5,2) belongs to $T$ and (5,0) belongs to $T$

2. (5,2) belongs to $T$ and (5,0) does not belong to $T$

3. (5,2) does not belong to $T$ and (5,0) belonsg to $T$

4. (5,2) does not belong to $T$ and (5,0) does not belong to $T$

The answer is 4 but I am not sure why it is 4, and what is the best approach to solve this kind of problem?

Thank you.

Answer 1

Use the definitions! Often mathematics is simply unfolding the definitions and statements, and see what is left.

Firstly, let's clear up what they are asking. They are asking whether $(5, 2) \in T$ or not, and whether $(5, 0)$ belongs to $T$.

$T$ is defined as the composition, so $$ T = \{(x, z) \mid x, z \in \mathbb{N} \land \exists y \in \mathbb{N} \colon (x, y) \in R \land (y, z) \in S\}. $$ That is, $T$ is the sets of ordered pairs $(x, z)$ such that there is a $y \in \mathbb{N}$ such that $x \mathrel{R} y$ and $y \mathrel{S} z$. In this case that means $x > y + 2$ and $y = z - 2$, equivalently $$ x > y + 2 = z - 2 + 2 = z. $$ So $x \mathrel{T} z$ (that is, $(x, z) \in T$) if and only if $x > z$.