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Step 1 of 4 4.051E Write the general Shannon's expansion theorem for a function Z with the input variables Write the other form or duality of Shannon's expansion theorem. Simplify the above two equations further to n-1 variable only if Use the switching axioms. In order to reduce the function by one literal, two cells in a group are necessary. It can be obtained only by two adjacent cells that differ by a single literal. It means that 2¹ cells are necessary. Step 2 of 4 Shannon's expansion theorem can be done by more than one variable. For example from the first equation expanding the function with respect to and gives as follows. For second equation expanding the function with respect to and Step 3 of 4 Simplify the above two equations further to n-2 variable only if Use the switching axioms. xy+xy=y In order to reduce the function by two literals, four cells in a group are necessary. It can be obtained only by four adjacent cells that differ by a two literal. It means that cells are necessary. Step 4 of 4 Similarly, when we expand the first and second equation with respect to three terms X1, X₂ and And to reduce the product or sum term by three literals we need eight (2³) adjacent cells. Hence by means of switching algebra and Shannon's theorem we can conclude that to reduce number of literals in the product term or sum term of the logical functions adjacent cells in K-map needs to be grouped.