I have two descriptions of two different relations and I need to determine whether they are functions. However, for some reason I have a really hard time interpreting what's going on here (esp. in (1)), maybe someone can help:
(1) $\{(a,b) \in \mathbb{Z} \times \mathbb{Z} | 7 | (b-a), -a -3 \leq b \leq -a +3\}$
(2) $\{(a,b) \in \mathbb{Z} \times \mathbb{Z} | b = |a|\}$
I'd be grateful for any tips and pointers!
Cheers!
Yes, both of them are functions. That's easy to see in the second case: that's simply the function $a\mapsto|a|$.
The first one is the function $f\colon\Bbb Z\longrightarrow\Bbb Z$ defined by$$f(a)=\text{only element of }\{a-3,a-2,\ldots,a+3\}\text{ which is a multiple of }7.$$Note that this makes sense, since $\{a-3,a-2,\ldots,a+3\}$ consists of $7$ consecutive integers, and therefore it has one and only one multiple of $7$. And, with this definition, $7\mid\bigl(f(a)-a\bigr)$ and $a-3\leqslant f(a)\leqslant a+3$.