Exam 12: A: Boolean Algebra

Find the sum-of-products expansion of the Boolean function f(x, y, z) that is 1 if and only if exactly two of the three variables have value 1.

Show that the Boolean function $F \text { given by } F ( x , y , z ) = x ( z + y z ) + y ( \overline { x z } x ) \text { simplifies to } x z + \bar { x } y \text {, by using }$ only the definition of a Boolean algebra.

mark each statement TRUE or FALSE. -The circuit diagrams for $x + \bar { x } y \text { and } y + x \bar { y } \text { produce the same output. }$

mark each statement TRUE or FALSE. - $\text { There are } n ^ { 2 } \text { minterms in the variables } x _ { 1 } , x _ { 2 } , \ldots , x _ { n } \text {. }$

mark each statement TRUE or FALSE. - $\overline { x \downarrow y } = \bar { x } \mid \bar { y }$

mark each statement TRUE or FALSE. -The circuit diagrams for $\overline { \bar { x } \bar { y } + \overline { x y } } \text { and } x + y \text { produce the same output }$

Write x + y as a sum-of-products in the variables x and y.

fill in the blanks. -There are ____ Boolean functions with 4 variables.

fill in the blanks. -Using $\downarrow \text { " for "nor," } ( x \downarrow x ) \downarrow ( y \downarrow y )$ can be written in terms of , +, and · as ____.

Find a Boolean function $F : \{ 0,1 \} ^ { 2 } \rightarrow \{ 0,1 \} \text { such that } F ( 0,0 ) = F ( 0,1 ) = F ( 1,1 ) = 1 \text { and } F ( 1,0 ) = 0$

mark each statement TRUE or FALSE. -When written as a sum of minterms in the variables x and $y , x + \bar { y } = x y + x \bar { y } + \bar { x } \bar { y }$

Construct a circuit using inverters, OR gates, and AND gates that gives an output of 1 if and only if three people on a committee do not all vote the same.

If $f ( w , x , y , z )$ = $( \bar { x } + y \bar { z } )$ + $( \bar { w } x )$ , find $f$ (1,0,1,1)

Give a reason for each step in the proof that x + x = x is true in Boolean algebras. Your reasons should come from the following: associative laws for addition and multiplication, commutative laws for addition and multiplication, distributive law for multiplication over addition and distributive law for addition over multiplication, identity laws, unit property, and zero property. $x = x + 0 = x + ( x \bar { x } ) = ( x + x ) ( x + \bar { x } ) = ( x + x ) \cdot 1 = 1 \cdot ( x + x ) = x + x$

Prove that the set of real numbers, with addition and multiplication of real numbers as + and · , negation as complementation, and the real numbers 0 and 1 as the 0 and the 1 respectively, is not a Boolean algebra.

Write $x ( y + 1 )$ as a sum-of-products in the variables x and y.

mark each statement TRUE or FALSE. - $\text { If } f ( z , y , z ) = x y z \text {, then } \overline { f ( z , y , z ) } = \bar { x } \bar { y } \bar { z } \text {. }$

Write x + z as a sum-of-products in the variables x, y, and z.

$\text { Let } F ( x , y , z ) = \bar { y } ( \bar { x } z ) + y x + y \bar { z }$ Draw a logic gate diagram for F .

Prove that F=G , where $\vec { F } ( x , y )$ = $( x + \bar { x } y ) \bar { y }$ and $G ( x , y ) = \overline { x + y }$