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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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It looks like you've made a mistake. Should it say $1 \le i \le m$ instead of just $\le i \le m$? – Fly by Night Oct 16 '12 at 15:09
I prefer the notation $\{(a_i,b_j) | 1 \leq i \leq m, \ \ 1 \leq j \leq n \}$. – copper.hat Oct 16 '12 at 15:12
@copper.hat That seems to be a different set. Let $a=(1,2)$ and $b=(1,2)$. In your notation $R=\{(1,1),(1,2),(2,1),(2,2)\}$ while in the OP's notation $R = \{1\times 1, 1 \times 2, 2\times 1, 2\times 2\} = \{1,2,4\}.$ – Fly by Night Oct 16 '12 at 15:18
@FlybyNight. In some of my books the notation $a_i\times b_j=(a_i,b_j)$ is used. – SomethingWitty Oct 16 '12 at 15:43
@SomethingWitty Gosh, really? That seems like terrible notation to me, in so many ways! I see why Copper Hat might prefer $(a_i,b_j)$. – Fly by Night Oct 16 '12 at 15:47
up vote 1 down vote accepted

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:

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

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