Question

given numberline A={a,b,c,d} and B={e,f,g,h} ,where a,b,c,d,e,f,g and h are naturals
question: how many will be the product of A and B ?

Answer 1

my opinion : multiply term by term

A×B= { a*e,a*f,a*g,a*h,b*e,b*f,b*g,b*h,c*e,c*f,c*g,c*h,d*e,d*f,d*g,d*h}

is this correct solved ?

Answer 2

@Vocaloid @Shadow When you have a chance

Answer 3

not really sure, I would think that's the right place to start

@hero @Angle

Answer 4

I feel like the question is phrased a little confusingly?

If the question is:
How many (of a, b, c, d, e, f, g, h) will be a product of A and B?
then the answer would be none, as "a" or "b" alone would never be a product of A and B by itself without an identity

If the question is:
How many products are there of A and B?
then you are correct
A×B= { a*e,a*f,a*g,a*h,b*e,b*f,b*g,b*h,c*e,c*f,c*g,c*h,d*e,d*f,d*g,d*h}
is the correct product and there are 16 elements

Answer 5

It could also be nice to note that the Natural Numbers are "closed" under multiplication. Which means that any product between two natural numbers is also a natural number.

Answer 6

It's called Cartesian Product:

If you're given two sets \(A\) and \(B\), the Cartesian product \(A × B\) is the set of all ordered pairs \((a, b)\) where \(a ∈ A\) and \(b ∈ B\). Products can be specified using set-builder notation. For example:
\(A \times B = \{(a,b) | a \in B \text{ and } b \in B\}\)

Simple Example:

\(A = {1,2}; B = {3,4}

\\A × B = \{1,2\} × \{3,4\} = \{(1,3), (1,4), (2,3), (2,4)\}
\\B × A = \{3,4\} × \{1,2\} = \{(3,1), (3,2), (4,1), (4,2)\}\)

Source: Wikipedia

Answer 7

A Visual Of The Same Concept:

Answer 8

ty all your help so my qustion is more much complicated bc. in reality

A={5,7,11,13} and B=A

so how can i getting A×A so that i need getting 5*5=25 and 5*7=35 so 5*11=55 and so much term by term ?

Answer 9

@jhonyy9 would you mind uploading the question as stated in the original document?

Answer 10

sorry @Hero this is my new idea togheter with my one colleg to getting in an easy way all prims greater than 3 and look that is right just we dont know again how we can getting these products starting with 5*5 and 5*7 and so much

so this is why i have asked it here in this way

Answer 11

I understand, but the problem is the language that you're using to express your question is not clear enough. The question you're posting needs to be stated in mathematical language that is clear enough to understand. Are you saying that the question you're posing is related to a concept other than Cartesian Product?

Answer 12

sorry and yes now this mean that you understand it right - yes we need getting these product different the Cartesian Product - so do you know any way to get these so how i have wrote it above ,please ?

Answer 13

so you're saying you want something like

{5*5, 5*7, 5*11, 5*13, 7*5, 7*7, 7*11, 7*13, 11*5, 11*7, 11*11, 11*13, 13*5, 13*7, 13*11, 13*13}
?

or

{5*5, 5*7, 5*11, 5*13, 7*7, 7*11, 7*13, 11*11, 11*13, 13*13}
?

or

something entirely different?

Answer 14

yes @Angle exactly

Answer 15

which one?
I gave two suggestions

Answer 16

ok the last one - sorry

Answer 17

every term multiplied by every term but just one time

Answer 18

bc. 5*7=7*5

Answer 19

"an easy way to get all primes greater than three". Not sure this is the way. And those prime products are not producing any prime numbered results btw.

Answer 20

@Hero please believe me we ,i togheter with my college have made the calcules and this is one right way getting every primes just we need subtracting from these number line every product of these terms how i have wrote it above and how have wrote @Angel in the last one case

Answer 21

The crossed out boxes are the ones you don't want

Answer 22

yes @Angle really this was my first idea too in this way but i hope so much that exist one clearly mathematics way getting these products using multiplication or any different way

Answer 23

or may dont exist so we need using this way how you made above

Answer 24

@Hero please do you understand it now my question ,problem please ?

Answer 25

I understand what @Angle is doing, but not anything you've said.

Answer 26

I also understand that Romanian and Hungarian are your native languages, but English is not.

Answer 27

ok ty . so how you think there exist any way in mathematics getting these products of these numberlines or just the way what have wrote @Angle above in this table ?

Answer 28

I do know that you can figure out amount of elements you get in the end as "n!"

however, I do not know of a short cut or formula for you to skip multiplying every element with every other element individually

Answer 29

@Angle using factorials - how ? please - with these numberlines

Answer 30

oh wait, sorry, It's not a factorial

I was referring to how
{5*5, 5*7, 5*11, 5*13, 7*7, 7*11, 7*13, 11*11, 11*13, 13*13}
gives you 10 terms

because
{5, 7, 11, 13} is 4 terms

and 4 + 3 + 2 + 1 = 10

Answer 31

ok. np. but please you @Angle and @Hero and indifferent everybody who has an idea what can helping in this case how we can getting these products - please reply me - thank you in advance all your help

Answer 32

@Angle posted the correct approach to this. You perform the cartesian product \(A \times B\) where \(B = A\) in this case so

\(A = \{5, 7, 11, 13\} \\ B = \{5, 7, 11, 13\}\)

Therefore
\(A \times B \\= \{5, 7, 11, 13\} \times \{5, 7, 11, 13\} \\=\{(5,5), (5,7), (5,11), (5,13),(7,7),(7,11),(7,13),(11,11),(11,13),(13,13)\}\)

Answer 33

ty @Hero so i think that finaly this will be the right way mathematicaly what using we can getting easy all primes

Answer 34

that is not the product

Answer 35

Yeah, it's not the entire product. I'm aware of this, but @jhonyy had some strict specifications for what he wanted the result to be.

Answer 36

we should use a different notation then

Answer 37

If you want to make a contribution to this, my all means. Be my guest.

Answer 38

I know of no standard notation what works for this particular, but we probably shouldn't use \[A\times B\]

Answer 39

'this particular problem'

Answer 40

I'm all eyes to see what you will recommend as far as correct notation.

Answer 41

\[\{(a,b)|a\in A,b\in B,a\leq b\}\]

Answer 42

I see. Yes, that's it.

Answer 43

That's the formula he's looking for I believe.