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Given a matroid, the IndependentSets command returns the independent sets of that matroid.

If the independent sets of this matroid haven't been computed before, they are computed by this command and stored for any future computations that use the independent sets.

$\mathrm{with}\left(\mathrm{Matroids}\right)\:$

$A≔\mathrm{Matrix}\left(\left[\left[1\,1\,1\,0\,0\,0\right]\,\left[0\,0\,0\,1\,1\,1\right]\,\left[1\,0\,0\,1\,0\,0\right]\,\left[0\,1\,0\,0\,1\,0\right]\,\left[0\,0\,1\,0\,0\,2\right]\right]\right)$

$A≔\left[\begin{array}{cccccc}1& 1& 1& 0& 0& 0\\ 0& 0& 0& 1& 1& 1\\ 1& 0& 0& 1& 0& 0\\ 0& 1& 0& 0& 1& 0\\ 0& 0& 1& 0& 0& 2\end{array}\right]$

$M≔\mathrm{Matroid}\left(A\right)$

$M≔⟨\begin{array}{c}\mathrm{th\ⅇ\; lin\ⅇar\; matroi\ⅆ\; whos\ⅇ\; groun\ⅆ\; s\ⅇt\; is\; th\ⅇ\; s\ⅇt\; of\; column\; v\ⅇctors\; of\; th\ⅇ\; matrix:}\\ \left[\begin{array}{cccccc}1& 1& 1& 0& 0& 0\\ 0& 0& 0& 1& 1& 1\\ 1& 0& 0& 1& 0& 0\\ 0& 1& 0& 0& 1& 0\\ 0& 0& 1& 0& 0& 2\end{array}\right]\end{array}⟩$

$\mathrm{IndependentSets}\left(M\right)$

$\left[\varnothing \,\left\{1\right\}\,\left\{2\right\}\,\left\{3\right\}\,\left\{4\right\}\,\left\{5\right\}\,\left\{6\right\}\,\left\{1\,2\right\}\,\left\{1\,3\right\}\,\left\{2\,3\right\}\,\left\{1\,4\right\}\,\left\{2\,4\right\}\,\left\{3\,4\right\}\,\left\{1\,5\right\}\,\left\{2\,5\right\}\,\left\{3\,5\right\}\,\left\{4\,5\right\}\,\left\{1\,6\right\}\,\left\{2\,6\right\}\,\left\{3\,6\right\}\,\left\{4\,6\right\}\,\left\{5\,6\right\}\,\left\{1\,2\,3\right\}\,\left\{1\,2\,4\right\}\,\left\{1\,3\,4\right\}\,\left\{2\,3\,4\right\}\,\left\{1\,2\,5\right\}\,\left\{1\,3\,5\right\}\,\left\{2\,3\,5\right\}\,\left\{1\,4\,5\right\}\,\left\{2\,4\,5\right\}\,\left\{3\,4\,5\right\}\,\left\{1\,2\,6\right\}\,\left\{1\,3\,6\right\}\,\left\{2\,3\,6\right\}\,\left\{1\,4\,6\right\}\,\left\{2\,4\,6\right\}\,\left\{3\,4\,6\right\}\,\left\{1\,5\,6\right\}\,\left\{2\,5\,6\right\}\,\left\{3\,5\,6\right\}\,\left\{4\,5\,6\right\}\,\left\{1\,2\,3\,4\right\}\,\left\{1\,2\,3\,5\right\}\,\left\{1\,3\,4\,5\right\}\,\left\{2\,3\,4\,5\right\}\,\left\{1\,2\,3\,6\right\}\,\left\{1\,2\,4\,6\right\}\,\left\{1\,3\,4\,6\right\}\,\left\{2\,3\,4\,6\right\}\,\left\{1\,2\,5\,6\right\}\,\left\{1\,3\,5\,6\right\}\,\left\{2\,3\,5\,6\right\}\,\left\{1\,4\,5\,6\right\}\,\left\{2\,4\,5\,6\right\}\,\left\{3\,4\,5\,6\right\}\,\left\{1\,2\,3\,4\,6\right\}\,\left\{1\,2\,3\,5\,6\right\}\,\left\{1\,3\,4\,5\,6\right\}\,\left\{2\,3\,4\,5\,6\right\}\right]$

James G. Oxley. Matroid Theory (Oxford Graduate Texts in Mathematics). New York: Oxford University Press. 2006.