Exam 12: Boolean Algebra

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

There are Boolean functions with 2 variables.

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

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

In questions mark each statement TRUE or FALSE. - $\text { Every Boolean function can be written using only the operators }^ - , + \text {, and } \cdot \text {. }$

Use the Quine-McCluskey method to simplify the Boolean expression $\bar { x } y z + \bar { x } y \bar { z } + \bar { x } \bar { y } z + \bar { x } \bar { y } \bar { z } + x y z$

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

When written as a product of maxterms (in the variables x and y ), $( x + y ) z =$$\underline{\quad\quad}$

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

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

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

A circuit is to be built that takes the numbers 0 through 9 as inputs (1 = 0001, 2 = 0010, . . . , 9 = 1001). Let $F ( w , x , y , z )$ be the Boolean function that produces an output of 1 if and only if the input is an even number. Find a Karnaugh map for F and use the map and don't care conditions to find a simple expression for F.

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

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

(a) Find a Boolean function $f : \{ 0,1 \} ^ { 3 } \rightarrow \{ 0,1 \}$ such that $f ( 1,1,0 ) = 1 , f ( 0,1,1 ) = 1 ,$ and $f ( x , y , z ) = 0$ otherwise. (b) Write $f$ using only . and $\underline{\quad\quad}$

Let $F ( x , y , z ) = \overline { ( \bar { y } z ) } ( x + \bar { x } y )$ . Show that F can be simplified to give $y + x \bar { z }$

Using can be written in terms of $- + , \text { and } \cdot \text { as }$$\underline{\quad\quad}$

Let F(x, y, z) = y (x z) + y x + y z. Use a Karnaugh map to simplify the function F .

Using PPP can be written in terms of PPP

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