$\mathcal{I}$</span> are called <em>independent</em>. </p><p>The <em>rank</em> of a matroid <span>$M=(S,\mathcal{I})$</span> is the size of a largest independent subset of <span>$S$</span>. An independent set of maximum rank is called a <em>basis</em>. </p><p>For <span>$X \subseteq S$</span>, the <em>rank</em> of <span>$X$</span> is the size of a largest independent subset of <span>$X$</span>.</p><p>See a matroid textbook or <a href="https://en.wikipedia.org/wiki/Matroid">online resource</a> for more detail