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• A minterm is the product of N distinct literals where each literal occurs exactly once.
• A maxterm is the sum of N distinct literals where each literal occurs exactly once.

For a two-variable expression, the minterms and maxterms are as follows

For a three-variable expression, the minterms and maxterms are as follows

This allows us to represent expressions in either Sum of Products or Product of Sums forms

The Sum of Products form represents an expression as a sum of minterms.

F(X, Y, ...) = Sum (a_k.m_k)

where a_k is 0 or 1 and m_k is a minterm.

To derive the Sum of Products form from a truth table, OR together all of the minterms which give a value of 1.

Consider the truth table

Here SOP is f(X.Y) = X.Y' + X.Y

The Product of Sums form represents an expression as a product of maxterms.

F(X, Y, .......) = Product (b_k + M_k), where b_k is 0 or 1 and M_k is a maxterm.

To derive the Product of Sums form from a truth table, AND together all of the maxterms which give a value of 0.

Consider the truth table from the previous example.

Here POS is F(X,Y) = (X+Y')

Give the expression represented by the following truth table in both Sum of Products and Product of Sums forms.

Conversion between the two forms is done by application of DeMorgans Laws.

As with any other form of algebra you have encountered, simplification of expressions can be performed with Boolean algebra.

Show that X.Y.Z' + X'.Y.Z' + Y.Z = Y

X.Y.Z' + X'.Y.Z' + Y.Z = Y.Z' + Y.Z = Y

Show that (X.Y' + Z).(X + Y).Z = X.Z + Y.Z

(X.Y' + Z).(X + Y).Z

= (X.Y' + Z.X + Y'.Z).Z

= X.Y'Z + Z.X + Y'.Z

= Z.(X.Y' + X + Y')

= Z.(X+Y')