Directed evolution requires the creation of genetic diversity and subsequent screening or selection for improved variants. For DNA mutagenesis, conventional site-directed methods implicitly utilize the Boolean AND operator (creating all mutations simultaneously), producing a combinatorial explosion in the number of genetic variants as the number of mutations increases. We introduce GeneORator, a novel strategy for creating DNA libraries based on the Boolean logical OR operator. Here, a single library is divided into many subsets, each containing different combinations of the desired mutations. Consequently, the effect of adding more mutations on the number of genetic combinations is additive (Boolean OR logic) and not exponential (AND logic). We demonstrate this strategy with large-scale mutagenesis studies, using monoamine oxidase-N (Aspergillus niger) as the exemplar target. First, we mutated every residue in the secondary structure-containing regions (276 out of a total 495 amino acids) to screen for improvements in kcat. Second, combinatorial OR-type libraries permitted screening of diverse mutation combinations in the enzyme active site to detect activity toward novel substrates. In both examples, OR-type libraries effectively reduced the number of variants searched up to 1010-fold, dramatically reducing the screening effort required to discover variants with improved and/or novel activity. Importantly, this approach enables the screening of a greater diversity of mutation combinations, accessing a larger area of a protein’s sequence space. OR-type libraries can be applied to any biological engineering objective requiring DNA mutagenesis, and the approach has wide ranging applications in, for example, enzyme engineering, antibody engineering, and synthetic biology.