Question 1

mark each statement TRUE or FALSE.
-$x x x + x y + y y = x y$

Accepted Answer

A) True 
 B)False

Question 2

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

Accepted Answer

A) True 
 B)False

Question 3

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

Accepted Answer

11ec7916_5586_c6a8_bdf9_f1cab22dec99_TB3

Question 4

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

Accepted Answer

A) True 
 B)False

Question 5

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

Accepted Answer

A) True 
 B)False

Question 6

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

Accepted Answer

A) True 
 B)False

Question 7

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

Accepted Answer

The answer of \(\text { Let } F ( x...

Question 8

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

Accepted Answer

A) True 
 B)False

Question 9

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

Accepted Answer

A) True 
 B)False

Question 10

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.

Accepted Answer

The answer of Using only the ﬁve properties associative laws,...

Question 11

mark each statement TRUE or FALSE.
-{+, ·} is a functionally complete set of operators.

Accepted Answer

A) True 
 B)False

Question 12

ﬁll in the blanks.
-There are ____ Boolean functions with 3 variables.

Accepted Answer

The answer of ﬁll in the blanks.
-There are ____ Boolean...

Question 13

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

Accepted Answer

The answer of If \(f ( w , x ,...

Question 14

ﬁll in the blanks.
-When written as a product of maxterms (in the variables x and y), $( x + y ) z =$ ____.

Accepted Answer

The answer of ﬁll in the blanks.
-When written as a...

Question 15

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

Accepted Answer

The answer of Show that the Boolean function \[F \text...

Question 16

Give a reason for each step in the proof that x + 1 = 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. $$1 = x + \bar { x } = x + \bar { x } \cdot 1 = ( x + \bar { x } ) ( x + 1 ) = 1 \cdot ( x + 1 ) = x + 1$$

Accepted Answer

Additive property of complemen

Question 17

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

Accepted Answer

A) True 
 B)False

Question 18

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

Accepted Answer

The answer of Write \(x y + x y ^...

Question 19

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. $$\begin{array} { l } 
x + x y = x \cdot 1 + x y = x ( y + \bar { y } ) + x y = ( x y + x \bar { y } ) + x y = x y + ( x y + x \bar { y } ) = ( x y + x y ) + x \bar { y } = x y + x \bar { y } = \
x ( y + \bar { y } ) = x \cdot 1 = x
\end{array}$$

Accepted Answer

Multiplicative property of 1, additive p

Question 20

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

Accepted Answer

The answer of If \(f ( w , x ,...

Exam 12: A: Boolean Algebra

mark each statement TRUE or FALSE. - xxx+xy+yy=xyx x x + x y + y y = x yxxx+xy+yy=xy

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

Use the Quine-McCluskey method to simplify the Boolean expression xˉyz+xˉyzˉ+xˉyˉz+xˉyˉzˉ+xyz\bar { x } y z + \bar { x } y \bar { z } + \bar { x } \bar { y } z + \bar { x } \bar { y } \bar { z } + x y zxˉyz+xˉyzˉ+xˉyˉ​z+xˉyˉ​zˉ+xyz

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

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

mark each statement TRUE or FALSE. - x+y=xˉyˉx + y = \bar { x } \quad \bar { y }x+y=xˉyˉ​

Let F(x,y,z)=(yˉz)(x+xˉy). \text { Let } F ( x , y , z ) = ( \bar { y } z ) ( x + \bar { x } y ) \text {. } Let F(x,y,z)=(yˉ​z)(x+xˉy). Show that F can be simpliﬁed to give y+xzˉy + x \bar { z }y+xzˉ

mark each statement TRUE or FALSE. - x∣y‾=x↓y\overline { x \mid y } = x \downarrow yx∣y​=x↓y

mark each statement TRUE or FALSE. -When written as a sum of minterms in the variables x and y,1=xy+xyˉ+xˉy+xˉyˉy , 1 = x y + x \bar { y } + \bar { x } y + \bar { x } \bar { y }y,1=xy+xyˉ​+xˉy+xˉyˉ​

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

mark each statement TRUE or FALSE. -{+, ·} is a functionally complete set of operators.

ﬁll in the blanks. -There are ____ Boolean functions with 3 variables.

If f(w,x,y,z)f ( w , x , y , z )f(w,x,y,z) = (xˉ+yzˉ)‾\overline{( \bar { x } + y \bar { z } )}(xˉ+yzˉ)​ + (wˉx)( \bar { w } x )(wˉx) , find f(0,0,0,0)f ( 0,0,0,0 )f(0,0,0,0)

ﬁll in the blanks. -When written as a product of maxterms (in the variables x and y), (x+y)z=( x + y ) z =(x+y)z= ____.

mark each statement TRUE or FALSE. - xˉyˉ=((x∣x)∣(y∣y))∣((x∣x)∣(y∣y))\bar { x } \bar { y } = ( ( x \mid x ) \mid ( y \mid y ) ) \mid ( ( x \mid x ) \mid ( y \mid y ) )xˉyˉ​=((x∣x)∣(y∣y))∣((x∣x)∣(y∣y))

Write xy+xyzˉx y + x y ^ { \bar { z } }xy+xyzˉ as a sum-of-products in the variables x, y, and z.

If f(w,x,y,z)f ( w , x , y , z )f(w,x,y,z) = (xˉ+yzˉ)\left( \bar { x } + y \bar { z } \right)(xˉ+yzˉ) + (wˉx)( \bar { w } x )(wˉx) , find fff (1,1,1,1) .

The Foundations: Logic and Proofs

A: the Foundations: Logic and Proofs

Basic Structures: Sets, Functions, Sequences, Sums, Matrices

A: Basic Structures: Sets, Functions, Sequences, Sums, Matrices

Algorithms

A: Algorithms

Number Theory and Cryptography

A: Number Theory and Cryptography

Induction and Recursion

A: Induction and Recursion

Counting

A: Counting

Discrete Probability

A: Discrete Probability

Advanced Counting Techniques

A: Advanced Counting Techniques

Relations

A: Relations

Graphs

A: Graphs

Trees

A: Trees

Boolean Algebra

Modeling Computation

A: Modeling Computation

Mathematics Problem Set: Set Theory, Number Theory, Combinatorics, and Boolean Algebra

Filters

mark each statement TRUE or FALSE. - $x x x + x y + y y = x y$

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

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

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

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

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

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

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

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

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.

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

ﬁll in the blanks. -When written as a product of maxterms (in the variables x and y), $( x + y ) z =$ ____.

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

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

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