Let S be a finite set, with |S|=n. If the bit matrix MR representing R contains exactly r entries that are 0, how many entries of MRbar are 0? Note: "MRbar" is supposed to be just like MR, which you should know has R as a subscript, but with a bar over R. Just in case that got confusing.
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If is a binary relation on a set , then can be represented by the logical matrix whose row and column indices index the elements of , such that the entries of are defined by:
Let . If is a binary relation on a set , then .
It follows from defenition of that . If the bit matrix representing contains exactly entries that are 0, then . Since , we conclude that the bit matrix representing contains exactly entries that are 0.