Take the 2-minute tour ×
Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It's 100% free, no registration required.

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)?

share|improve this question
    
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
show 1 more comment

2 Answers 2

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 $b$.

share|improve this answer
add comment

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:

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

share|improve this answer
add comment

Your Answer

 
discard

By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.