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I am getting confused with this notation:

$R =\{x | x=a_i\times b_j;1\leq i\leq m \text{ and } 1\leq j\leq n\}$ where $a$ and $b$ are vectors of length $m$ and $n$ resp.

What does this mean? Does this mean x is the set of the products of the first element of a with all the elements of b and then the second element with all of b and so on?

(a1b1, a1b2, ..., a1bn,a2b1,a2b2...ambn)?

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    Munkres (Topology, A First Course) uses $a \times b$ to mean an ordered pair (more usually $(a,b)$). It could also be the set of (multiplicative) products.2012-10-16

2 Answers 2

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What you write down seems to be correct, although $R$ is not ordered. It is the set of all products of one coordinate of $a$ with one coordinate of $b$.

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The set $R$ is given by the set of numbers formed by multiplying all of the different components of $\underline{a}$ with all of the different components of $\underline{b}.$ If $\underline{a} = (a_1,\ldots,a_m)$ and $\underline{b} = (b_1,\ldots,b_n),$ then $R$ is made up of all of the products $a_i \times b_j$ where $i$ can range from $1$ to $m$ and $j$ can range from $1$ to $n$. In other words, $R$ is all products $a_i \times b_j$ where $1 \le i \le m$ and $1 \le j \le n.$ Thus:

$R = \{a_1b_1,\ldots,a_1b_n,a_2b_1,\ldots,a_2b_n,\ldots,a_mb_1,\ldots,a_mb_n\} \, . $