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Let $A$ be the set $\{ 1, 2, 3, 4\}$. Which ordered pairs are in the relation $R = \{ (a, b) | a\text{ divides }b\}$?

Solution: Because $(a, b)$ is in $R$ if and only if $a$ and $b$ are positive integers not exceeding $4$ such that $a$ divides $b$, we see that

$$R = \{ (1, 1), (1, 2), (1, 3), (1, 4), (2, 2), (2, 4), (3, 3), (4, 4)\} .$$

Does a divides $b$ means $\exists c \in \mathbb{Z} $ such that $c\cdot a = b $? In other words, $b$ is divisible by $a$, in terms of integer?

2 Answers 2

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It means elements of range (b) should be divisible by elements of domain (a).

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Yes, it means just that, usual divisibility in the integers or natural numbers, the distinction does not matter in this case.

This is by far the most natural interpretation and matches the explicit description of $R$.

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    so in this case, (3,1), for example, is wrong in this relation, because c x 3 =1, $c = \frac{1}{3}$ and c must be a fraction, not an integer? – 2017-01-21
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    @ quid so in this case, (3,1), for example, is wrong in this relation, because c x 3 =1, $c = \frac{1}{3}$ and c must be a fraction, not an integer? – 2017-01-21
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    Yes, $(3,1)$ is not in the relation. As $3$ does not divide $1$. Yet $(1,3)$ is in the relation as $1$ divides $3$. – 2017-01-21