Chapter 12: Q17E (page 842)
\((a)\)Six variables contain\(64\) cells
\((b)\)There are \(6\) adjacent cells
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Deal with the Boolean algebra \(\{ 0,1\} \) with addition, multiplication, and complement defined at the beginning of this section. In each case, use a table as in Example \(8\).
Verify the zero property.
The Boolean operator \( \oplus \), called the \(XOR\) operator, is defined by \(1 \oplus 1{\bf{ = }}0,1 \oplus 0{\bf{ = }}1,0 \oplus 1{\bf{ = }}1\), and \(0 \oplus 0{\bf{ = }}0\).
Construct a circuit that computes the product of the two-bitintegers \({{\bf{(}}{{\bf{x}}_{\bf{1}}}{{\bf{x}}_{\bf{o}}}{\bf{)}}_{\bf{2}}}\)and\({{\bf{(}}{{\bf{y}}_{\bf{1}}}{{\bf{y}}_{\bf{o}}}{\bf{)}}_{\bf{2}}}\).The circuit should have four output bits for the bits in the product. Two gates that are often used in circuits are NAND and NOR gates. When NAND or NOR gates are used to represent circuits, no other types of gates are needed. The notation for these gates is as follows:
Show that each of these identities holds.
\({\bf{a)}}\)\({\bf{x}} \odot {\bf{x = 1}}\)
\({\bf{b)}}\)\({\bf{x}} \odot {\bf{\bar x = 0}}\)
\({\bf{c)}}\)\({\bf{x}} \odot {\bf{y = y}} \odot {\bf{x}}\)
Show that if \({\bf{F, G}}\), and \({\bf{H}}\) are Boolean functions of degree \({\bf{n}}\), then \({\bf{F + G}} \le {\bf{H}}\) if and only if \({\bf{F}} \le {\bf{H}}\) and \({\bf{G}} \le {\bf{H}}\).
Show that \({\bf{F}}\left( {{\bf{x, y, z}}} \right){\bf{ = x y + x z + y z}}\) has the value \(1\) if and only if at least two of the variables \({\bf{x, y}}\), and \({\bf{z}}\) have the value \(1\) .
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