The units of a cartesian product?

In summary, to find all units in Z12 X Z6 and their inverses, the conversation suggests taking the cartesian product of the units in Z12 and Z6. This would result in eight pairs: 1,1 1,5 11,1 11,5 5,1 5,5 7,1 and 7,5. To fully understand this concept, it is recommended to prove that the units of the product are precisely the pairs of units, which is a simple task of writing out the definitions.
  • #1
chuy52506
77
0
Find all units in [tex]Z[/tex]12 X [tex]Z[/tex]6 and their inverses.

What i did was find the units of Z12 which are 1,11,5,7 then the ones of Z6 which are 1,5 and take the cartesian product of those two sets?
 
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  • #2
Yes, that would be correct!
 
  • #3
So there would be eight?
1,1 1,5 11,1 11,5 5,1 5,5 7,1 and 7,5?
 
  • #4
Instead of asking this, why don't you try to prove that the units of the product are precisely the pairs of units?
 
  • #5
Because it helps to do an example before. So is this correct?
 
  • #6
micromass already answered that question. If you're still not entirely sure, my advice stands: prove it! (it's quite easy; it's a matter of writing out the definitions)
 

1. What are the units of a cartesian product and how are they calculated?

The units of a cartesian product are determined by multiplying the number of units in each set involved. For example, if set A has 3 units and set B has 4 units, the cartesian product of A and B would have 3x4=12 units.

2. Can the units of a cartesian product be negative or zero?

No, the units of a cartesian product cannot be negative or zero. The number of units in a cartesian product represents the total number of combinations between two sets, and negative or zero values do not make sense in this context.

3. How does the order of sets affect the units in a cartesian product?

The order of sets does not affect the units in a cartesian product. The number of units in a cartesian product is only determined by the number of units in each set, not the order in which they are multiplied.

4. Can a cartesian product have an infinite number of units?

Yes, a cartesian product can have an infinite number of units if at least one of the sets involved has an infinite number of units. For example, the cartesian product of an infinite set and a finite set would have an infinite number of units.

5. How is the concept of units in a cartesian product related to the concept of dimensions?

The number of units in a cartesian product is related to the concept of dimensions. In a cartesian product of two sets, the number of units represents the number of dimensions in the resulting space. For example, a cartesian product of two sets with 3 and 4 units respectively would result in a 3-dimensional space with 4 points in each dimension.

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