Matrices of Zeros and Ones

Let A be a. matrix of m rows and n columns and let the entries of A be the integers 0 and 1. We call such a matrix a (0, 1) -matrix of size m by n. The 2 (0, 1)-matrices of size m by n play a fundamental role in a wide variety of combinatorial investigations. One of the chief reasons for this is the following. Let X be a set of n elements Xi, X2, • • • , xn and let Xi, X2, • • • , Xm be m subsets of X. Let dij= 1 if Xj is a member of Xi and let 0,7 = 0 if Xj is not a member of Xi. The a»/s yield a (0, 1)-matrix 4 = [a»y] of size w by w called the incidence matrix for the subsets Xi, X2, • • • , Xm of X. The l 's in row i oî A specify the elements that belong to set Xi and the l 's in column j of A specify the sets that contain element x3-. The matrix A characterizes the m subsets Xi, X2, • • • , Xm of the set X. Let A be a (0, 1)-matrix of size m by n. het the sum of row i of A be denoted by r» and let the sum of column j of 4̂ be denoted by Sj. We call