Write a boolean expression for the following statement: ‘‘Z is TRUE if no more than two of W, X, and Y are TRUE, otherwise Z is FALSE.’’ Consider the statement: ‘‘Z is TRUE if at least two of W, X, and Y are TRUE, otherwise Z is FALSE.’’

 

a) Write a boolean expression for this statement.

b) Write a truth table for the function Z.

c) Implement Z using only AND, OR, and NOT gates. The inputs W, X, and Y are available. Each gate may have an arbitrary number of inputs. (Hint: A sum-of-products representation of the boolean expression will facilitate this implementation.)

d) Implement Z using only AND, OR, and NOT gates. Each gate may have no more than two inputs. As before, the inputs W, X, and Y are available.

e) Implement Z using only NAND and NOR gates. (Hint: a NAND gate or a NOR gate with its inputs tied together behaves like an inverter.)

f) Implement Z using only NAND gates. (Hint: Use De Morgan’s laws.)

g) Implement Z using only NOR gates. (Hint: Use De Morgan’s laws.)

h) Repeat part (d) and attempt to minimize the number of gates used.

i) Repeat part (d) and attempt to minimize the number of gates used, assuming that the inputs are available both in their true and complement forms. In other words, assume that in addition to W, X, and Y, the inputs W, X, and Y, are also available.