Exam 12: Boolean Algebra

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

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

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

When written as a sum of minterms (in the variables x and y ), $x + \bar { x } y =$$\underline{\quad\quad}$

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

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

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

Give a reason for each step in the proof that x + xy = 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, zero property, and idempotent laws. $x+x y=x \cdot 1+x y=x(y+\bar{y})+x y=(x y+x \bar{y})+x y=x y+(x y+x \bar{y})=(x y+x y)+x \bar{y}=x y+x \bar{y}=$$x(y+\bar{y})=x \cdot 1=x .$

Using only the five properties associative laws, commutative laws, distributive laws, identity laws, and com- plement laws, prove that x x = x is true in all Boolean algebras.

Use a Karnaugh map to minimize the sum-of-products expression $x y z + x \bar { y } z + \bar { x } \bar { y } z + x \bar { y } \bar { z } + \bar { x } y z + \bar { x } \bar { y } \bar { z }$

Find the sum-of-products expansion of the Boolean function $f ( x , y )$ that is 1 if and only if either $x = 0$ and $y = 1 \text {, or } x = 1 \text { and } y = 0 \text {. }$

Write 1 as a sum-of-products in the variables x and y.

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

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

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

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. $x = x + 0 = x + ( x \bar { x } ) = ( x + x ) ( x + \bar { x } ) = ( x + x ) \cdot 1 = 1 \cdot ( x + x ) = x + x$

There are Boolean functions with 3 variables.

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

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

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