Exam 12: Boolean Algebra

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

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

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

In questions determine whether the statement is TRUE or FALSE. Assume that x, y, and z represent Boolean variables. -x + x y + x = x.

Procomplemenve that thetation,setandof realthenurealmbners,umwithbers 0additionand 1 aands them0ultiplicationand the 1 respof realectivnelyum, bisersnotasa+Boandolean· , negationalgebra. as

Write $x y + x y \bar { z }$ as a sum-of-products in the variables $x , y , \text { and } z$

In questions mark each statement TRUE or FALSE. - $x + y = \bar { x } \mid \bar { y }$

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

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.

In questions determine whether the statement is TRUE or FALSE. Assume that x, y, and z represent Boolean variables. -(x x + 1) = (x + 1)(x + 1).

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

Using only the five properties associative laws, commutative laws, distributive laws, identity laws, and complement laws, prove that $x + 1 = 1$ is true in all Boolean algebras.

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

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

In questions determine whether the statement is TRUE or FALSE. Assume that x, y, and z represent Boolean variables. - $\overline { y z + \bar { x } } = \overline { y z } x$

In questions mark each statement TRUE or FALSE. - $\text { When written as a sum of minterms in the variables } x \text { and } y , 1 = x y + x \bar { y } + \bar { x } y + \bar { x } \bar { y } \text {. }$

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.