More complicated sets

How do you form the Cartesian product of a list with itself? In some cases, it is better to use ``indexed sets''. To take the $2nd$ power of $\{2,4,8\}$ with itself and then select the $3rd$ element: first create this as a set, then use the Subsequences( ,2); command, then convert that to an ``indexed set'' using the SetToIndexedSet command, as follows.

> Set1:={2^i:i in [1..3]};          
> Set1_2:=Subsequences(Set1,2);
> Set1_2_2:=SetToIndexedSet(Set1_2);
> Set1_2_2[3];
[ 2, 8 ]

We could have entered $\{2,4,8\}$ as an indexed set by typing
Set1:={@2^i:i in [1..3]@}; but then MAGMA would've objected to the next line since Subsequences is not defined for indexed sets.

To add an element to a sequence or set you can use the Include command. For example,

> newset:=Include(Set1_2_2,[333]);
> newset;
{@
    [ 8, 2 ],
    [ 4, 2 ],
    [ 2, 8 ],
    [ 8, 4 ],
    [ 4, 4 ],
    [ 8, 8 ],
    [ 2, 2 ],
    [ 4, 8 ],
    [ 2, 4 ],
    [ 333 ]
@}

Suppose you want to select a subset of a set having a certain property, how do you do this? Take, for example, the above cartesian product $\{(i,j)\ \vert\ i,j\in \{2,4,8\}\}$, and find all the elements which add to 6.

> Set1_2_2;
{@
    [ 8, 2 ],
    [ 4, 2 ],
    [ 2, 8 ],
    [ 8, 4 ],
    [ 4, 4 ],
    [ 8, 8 ],
    [ 2, 2 ],
    [ 4, 8 ],
    [ 2, 4 ]
@}
> L:=[ x : x in Set1_2_2 | x[1]+x[2] eq 6];
> L;
[
    [ 4, 2 ],
    [ 2, 4 ]
]