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Disjunct matrix
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<p>In mathematics, a <a href="/facts/Logical_matrix/pS4BnBzj">logical matrix</a> may be described as <i>d</i>-disjunct and/or <i>d</i>-separable. These concepts play a pivotal role in the mathematical area of non-adaptive <a href="/facts/Group_testing/8wLh7fmq">group testing</a>. </p><p>In the mathematical literature, <i>d</i>-disjunct matrices may also be called super-imposed codes or <i>d</i>-cover-free families. </p><p>According to Chen and Hwang (2006), </p> <ul><li>A matrix is said to be <i>d</i>-separable if no two sets of <i>d</i> columns have the same boolean sum.</li> <li>A matrix is said to be d ¯ {\displaystyle {\overline {d}}} -separable (that's <i>d</i> with an overline) if no two sets of <i>d</i>-or-fewer columns have the same boolean sum.</li> <li>A matrix is said to be <i>d</i>-disjunct if no set of <i>d</i> columns has a boolean sum which is a superset of any other single column.</li></ul> <p>The following relationships are "well-known": </p> <ul><li>Every d + 1 ¯ {\displaystyle {\overline {d+1}}} -separable matrix is also d {\displaystyle d} -disjunct.</li> <li>Every d {\displaystyle d} -disjunct matrix is also d ¯ {\displaystyle {\overline {d}}} -separable.</li> <li>Every d ¯ {\displaystyle {\overline {d}}} -separable matrix is also d {\displaystyle d} -separable (by definition).</li></ul>