Given a matroid $M$ and a subset $S$ of its ground set $E$, the contraction $M/S$ of $M$ by $S$ is a matroid whose ground set is $E\setminus S$.

The simplest description of $M/S$ is via its rank function. Let $r$ be the rank function of $M$, then the rank function $R$ of $M/S$ is given by:

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

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

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

$M≔\mathrm{Matroid}\left(A\right)$

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

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

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

$\mathrm{M2}≔\mathrm{Contraction}\left(M\,\left\{1\,2\right\}\right)$

$\mathrm{M2}≔⟨\mathrm{a\; matroi\ⅆ\; on\; 4\; \ⅇl\ⅇm\ⅇnts\; with\; 1\; circuit}⟩$

$\mathrm{IndependentSets}\left(\mathrm{M2}\right)$

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

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