Understanding relation R on sets of integers

Question

Let = ℤ and be the relation defined by = ℤ × ℤ − {(, ) | ∈ }.

a. Identify the relation , i.e. give the name or symbol that is in common usage for this relation.

b. Find ^2 and prove your answer.

Answer 1

You need to understand what definition of a "relationship" is and what it means.

Let's get colloquial.

We can say something simple like "$a$ and $b$ are related if $b$ is two times $a$". So $7$ and $14$ are related and $46$ and $92$ are related but $8$ and $15$ are not related and so on. Also $14$ and $7$ are not related because the relationship is only one way. The second number must be twice the first but not vice versa.

So how do we express the idea mathematically? Not just how to express the idea that one set of $b = 2a$ but the idea of the relationship?

Well, sets. $X \times Y$ is the set of all pairs $(x,y)$. So we want all the pairs $(x,y)$ in $\mathbb Z \times \mathbb Z$ where $x$ is related to $y$.

In this case we want $R\subset \mathbb Z \times \mathbb Z$ where $R = \{(a,b)| b = 2*a\}$.

In other words $R$ is the set of all pairs of integers, where the integers are related in a certain way. In math a "RELATION" is a set of ordered pairs where the pair have a certain condition.

In this case the relation is that $R = \{(a,b)| (a,b) \not \in \{(n,n)\}\}$ which can be written as $R = \mathbb Z \times \mathbb Z \setminus \{(n,n)|n \in \mathbb Z\}$.

So $a$ and $b$ are related if $(a,b)$ is NOT in $\{(n,n)|n \in \mathbb Z\}$.

So what is a way of saying that in simple english?

Hint: maybe first would be, how would you describe the relation between $a$ and $b$ if $(a,b)$ WERE in $\{(n,n)|n \in \mathbb Z\}$? So how would you describe it if they were not?