Exam 12: Boolean Algebra

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

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Question 1

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We construct the following K-map. $We construct the following K-map. This gives us the expansion \bar { y } + z$ This gives us the expansion $\bar { y } + z$

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

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Question 2

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Let $x , y$ , and $z$ represent the votes of the three people on the committee with a variable taking the value 1 if the vote is affirmative and the value 0 if the vote is negative. The circuit should produce an output of 1 if and only if not all three of the variables have the same value. The function $f ( x , y , z ) = \overline { ( x y z + \bar { x } \bar { y } \bar { z } ) }$ gives this output. Hence we can use the following circuit. $Let x , y , and z represent the votes of the three people on the committee with a variable taking the value 1 if the vote is affirmative and the value 0 if the vote is negative. The circuit should produce an output of 1 if and only if not all three of the variables have the same value. The function f ( x , y , z ) = \overline { ( x y z + \bar { x } \bar { y } \bar { z } ) } gives this output. Hence we can use the following circuit.$

What is the output of the following circuit?

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Question 3

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The output of the circuit is $\bar { x } y + x \bar { z }$

Prove or disprove that $x + x y + x y z = x$ whenever x, y, and z are Boolean variables.

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Question 4

How many diﬀerent Boolean functions are there of degree 3?

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Question 5

What is the value of the Boolean function $f ( x , y , z ) = ( \bar { x } + \bar { y } ) z + x y z \text { when } x = 1 , y = 0 \text { and } z = 1 ?$

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Question 6

Prove or disprove that x y + y = y whenever x and y are Boolean variables.

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Question 7

Use a K-map to minimize the sum-of-products expansion x y z + x y z + x y z + x y z.

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Question 8

Find a Boolean function f(x, y, z) that has the value 1 if and only if exactly two of x, y, and z have the value 1.

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Question 9

Is the set of operators {+, ·} functionally complete? Justify your answer.

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Question 10

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

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Question 11

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

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

What is the output of the following circuit?

Prove or disprove that $x + x y + x y z = x$ whenever x, y, and z are Boolean variables.

How many diﬀerent Boolean functions are there of degree 3?

What is the value of the Boolean function $f ( x , y , z ) = ( \bar { x } + \bar { y } ) z + x y z \text { when } x = 1 , y = 0 \text { and } z = 1 ?$

Prove or disprove that x y + y = y whenever x and y are Boolean variables.

Use a K-map to minimize the sum-of-products expansion x y z + x y z + x y z + x y z.

Find a Boolean function f(x, y, z) that has the value 1 if and only if exactly two of x, y, and z have the value 1.

Is the set of operators {+, ·} functionally complete? Justify your answer.

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

A: Boolean Algebra

Modeling Computation

A: Modeling Computation

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

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Exam 12: Boolean Algebra

Use a K-map to minimize the sum-of-products expansion xyz+xyˉz+xˉyˉz+xyˉzˉ+xˉyz+xˉyˉzˉx 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 }xyz+xyˉ​z+xˉyˉ​z+xyˉ​zˉ+xˉyz+xˉyˉ​zˉ

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

What is the output of the following circuit?

Prove or disprove that x+xy+xyz=xx + x y + x y z = xx+xy+xyz=x whenever x, y, and z are Boolean variables.

How many diﬀerent Boolean functions are there of degree 3?

What is the value of the Boolean function f(x,y,z)=(xˉ+yˉ)z+xyz when x=1,y=0 and z=1?f ( x , y , z ) = ( \bar { x } + \bar { y } ) z + x y z \text { when } x = 1 , y = 0 \text { and } z = 1 ?f(x,y,z)=(xˉ+yˉ​)z+xyz when x=1,y=0 and z=1?

Prove or disprove that x y + y = y whenever x and y are Boolean variables.

Use a K-map to minimize the sum-of-products expansion x y z + x y z + x y z + x y z.

Find a Boolean function f(x, y, z) that has the value 1 if and only if exactly two of x, y, and z have the value 1.

Is the set of operators {+, ·} functionally complete? Justify your answer.

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

A: Boolean Algebra

Modeling Computation

A: Modeling Computation

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

Filters

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

Prove or disprove that $x + x y + x y z = x$ whenever x, y, and z are Boolean variables.

What is the value of the Boolean function $f ( x , y , z ) = ( \bar { x } + \bar { y } ) z + x y z \text { when } x = 1 , y = 0 \text { and } z = 1 ?$