Boolean Algebra

by | Oct 17, 2023 | mathematics

Part 1: Boolean Algebra:

1. Write x ⊕ y in disjunctive normal form and conjunctive normal form. Using of these forms, simplify these expressions:
a) x ⊕ 0
b) x ⊕ 1
c) x ⊕ x
d) x ⊕ x ̅

2. Find the sum-of-products expansions of the Boolean function F(x, y, z) that equals 1 if and only if:
a) x = 0.
b) xy = 0.
c) x + y = 0.
d) xyz = 0

3. Express each of these Boolean functions using the operators ⋅ and −.
a) x + y + z
b) x + y ̅(x ̅ + z)
c) (x + y ̅ ) ̅
d) x ̅(x + y ̅ + z ̅)

Part 2: Sets:

1. For each of these pairs of sets, determine whether the first is a subset of the second, the second is a subset of the first, or neither is a subset of the other.
a) the set of people who speak English, the set of people who speak English with an Australian accent
b) the set of fruits, the set of citrus fruits
c) the set of students studying discrete mathematics, the set of students studying data structures
d) the set of students in the college of computing and IT, the set of students at UDST

2. Determine whether these statements are true or false.
a) ∅ ∈ {∅}
b) ∅ ∈ {∅, {∅}}
c) {∅} ∈ {∅}
d) {∅} ∈ {{∅}}
e) {∅} ⊂ {∅, {∅}}
f ) {{∅}} ⊂ {∅, {∅}}
g) {{∅}} ⊂ {∅, {{∅}}}
h) ∅ ⊂ {{∅}, {{∅}}}

3. Can you conclude (with a proof) that A = B if A, B, and C are sets such that
a) A ∪ C = B ∪ C?
b) A ∩ C = B ∩ C?
c) A ∪ C = B ∪ C and A ∩ C = B ∩ C?

4. Sets in python
a) Write a code to generate the set setA of positive integers less than 1000 which last two digits are divisible by 43 (example 186 is in the set because 86 is divisible by 43)
b) Write a code to generate the set setB of positive integers less than 1000 which first two digits are divisible by 24 (example 725 is in the set because 72 is divisible by 24)
c) Write a code to compute setC the union of setA and setB
d) Write a code to compute setD the intersection of setA and setB
e) Check using a python code that |setA|+|setB|=|setC|+|setD|
f) Write a function cartesian that has two sets as input and returns their cartesian product as output (without importing any library)!
g) Use your function to find the cartesian product between setD and the set of colors on a traffic light.

 

 

Sample solution

Dante Alighieri played a critical role in the literature world through his poem Divine Comedy that was written in the 14th century. The poem contains Inferno, Purgatorio, and Paradiso. The Inferno is a description of the nine circles of torment that are found on the earth. It depicts the realms of the people that have gone against the spiritual values and who, instead, have chosen bestial appetite, violence, or fraud and malice. The nine circles of hell are limbo, lust, gluttony, greed and wrath. Others are heresy, violence, fraud, and treachery. The purpose of this paper is to examine the Dante’s Inferno in the perspective of its portrayal of God’s image and the justification of hell. 

In this epic poem, God is portrayed as a super being guilty of multiple weaknesses including being egotistic, unjust, and hypocritical. Dante, in this poem, depicts God as being more human than divine by challenging God’s omnipotence. Additionally, the manner in which Dante describes Hell is in full contradiction to the morals of God as written in the Bible. When god arranges Hell to flatter Himself, He commits egotism, a sin that is common among human beings (Cheney, 2016). The weakness is depicted in Limbo and on the Gate of Hell where, for instance, God sends those who do not worship Him to Hell. This implies that failure to worship Him is a sin.


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God is also depicted as lacking justice in His actions thus removing the godly image. The injustice is portrayed by the manner in which the sodomites and opportunists are treated. The opportunists are subjected to banner chasing in their lives after death followed by being stung by insects and maggots. They are known to having done neither good nor bad during their lifetimes and, therefore, justice could have demanded that they be granted a neutral punishment having lived a neutral life. The sodomites are also punished unfairly by God when Brunetto Lattini is condemned to hell despite being a good leader (Babor, T. F., McGovern, T., & Robaina, K. (2017). While he commited sodomy, God chooses to ignore all the other good deeds that Brunetto did.

Finally, God is also portrayed as being hypocritical in His actions, a sin that further diminishes His godliness and makes Him more human. A case in point is when God condemns the sin of egotism and goes ahead to commit it repeatedly. Proverbs 29:23 states that “arrogance will bring your downfall, but if you are humble, you will be respected.” When Slattery condemns Dante’s human state as being weak, doubtful, and limited, he is proving God’s hypocrisy because He is also human (Verdicchio, 2015). The actions of God in Hell as portrayed by Dante are inconsistent with the Biblical literature. Both Dante and God are prone to making mistakes, something common among human beings thus making God more human.

To wrap it up, Dante portrays God is more human since He commits the same sins that humans commit: egotism, hypocrisy, and injustice. Hell is justified as being a destination for victims of the mistakes committed by God. The Hell is presented as being a totally different place as compared to what is written about it in the Bible. As a result, reading through the text gives an image of God who is prone to the very mistakes common to humans thus ripping Him off His lofty status of divine and, instead, making Him a mere human. Whether or not Dante did it intentionally is subject to debate but one thing is clear in the poem: the misconstrued notion of God is revealed to future generations.

 

References

Babor, T. F., McGovern, T., & Robaina, K. (2017). Dante’s inferno: Seven deadly sins in scientific publishing and how to avoid them. Addiction Science: A Guide for the Perplexed, 267.

Cheney, L. D. G. (2016). Illustrations for Dante’s Inferno: A Comparative Study of Sandro Botticelli, Giovanni Stradano, and Federico Zuccaro. Cultural and Religious Studies, 4(8), 487.

Verdicchio, M. (2015). Irony and Desire in Dante’s” Inferno” 27. Italica, 285-297.

Sample Answer

Sample Answer

 

Part 1: Boolean Algebra:

1. Write x ⊕ y in disjunctive normal form and conjunctive normal form. Using of these forms, simplify these expressions:
a) x ⊕ 0
b) x ⊕ 1
c) x ⊕ x
d) x ⊕ x ̅

2. Find the sum-of-products expansions of the Boolean function F(x, y, z) that equals 1 if and only if:
a) x = 0.
b) xy = 0.
c) x + y = 0.
d) xyz = 0

3. Express each of these Boolean functions using the operators ⋅ and −.
a) x + y + z
b) x + y ̅(x ̅ + z)
c) (x + y ̅ ) ̅
d) x ̅(x + y ̅ + z ̅)

Part 2: Sets:

1. For each of these pairs of sets, determine whether the first is a subset of the second, the second is a subset of the first, or neither is a subset of the other.
a) the set of people who speak English, the set of people who speak English with an Australian accent
b) the set of fruits, the set of citrus fruits
c) the set of students studying discrete mathematics, the set of students studying data structures
d) the set of students in the college of computing and IT, the set of students at UDST

2. Determine whether these statements are true or false.
a) ∅ ∈ {∅}
b) ∅ ∈ {∅, {∅}}
c) {∅} ∈ {∅}
d) {∅} ∈ {{∅}}
e) {∅} ⊂ {∅, {∅}}
f ) {{∅}} ⊂ {∅, {∅}}
g) {{∅}} ⊂ {∅, {{∅}}}
h) ∅ ⊂ {{∅}, {{∅}}}

3. Can you conclude (with a proof) that A = B if A, B, and C are sets such that
a) A ∪ C = B ∪ C?
b) A ∩ C = B ∩ C?
c) A ∪ C = B ∪ C and A ∩ C = B ∩ C?

4. Sets in python
a) Write a code to generate the set setA of positive integers less than 1000 which last two digits are divisible by 43 (example 186 is in the set because 86 is divisible by 43)
b) Write a code to generate the set setB of positive integers less than 1000 which first two digits are divisible by 24 (example 725 is in the set because 72 is divisible by 24)
c) Write a code to compute setC the union of setA and setB
d) Write a code to compute setD the intersection of setA and setB
e) Check using a python code that |setA|+|setB|=|setC|+|setD|
f) Write a function cartesian that has two sets as input and returns their cartesian product as output (without importing any library)!
g) Use your function to find the cartesian product between setD and the set of colors on a traffic light.
Part 1: Boolean Algebra
Disjunctive Normal Form (DNF) and Conjunctive Normal Form (CNF) of x ⊕ y:
DNF: (x ∧ ¬y) ∨ (¬x ∧ y) CNF: (x ∨ y) ∧ (¬x ∨ ¬y)

a) x ⊕ 0: DNF: (x ∧ ¬0) ∨ (¬x ∧ 0) = x ∧ ¬x = 0 CNF: (x ∨ 0) ∧ (¬x ∨ ¬0) = x ∧ ¬x = 0

b) x ⊕ 1: DNF: (x ∧ ¬1) ∨ (¬x ∧ 1) = ¬x ∨ x = 1 CNF: (x ∨ 1) ∧ (¬x ∨ ¬1) = 1

c) x ⊕ x: DNF: (x ∧ ¬x) ∨ (¬x ∧ x) = 0 ∨ 0 = 0 CNF: (x ∨ x) ∧ (¬x ∨ ¬x) = x ∧ ¬x = 0

d) x ⊕ ¬x: DNF: (x ∧ ¬¬x) ∨ (¬x ∧ ¬x) = x ∨ ¬x CNF: (x ∨ x) ∧ (¬x ∨ ¬x) = x ∧ ¬x

Sum-of-Products expansions of F(x, y, z):
a) F(x, y, z) = x’ + y + z

b) F(x, y, z) = xy’z’

c) F(x, y, z) = xyz’

d) F(x, y, z) = xy’z + xyz’

Expressing Boolean functions using ⋅ and −:
a) x + y + z: Using ⋅: x ⋅ y ⋅ z Using −: (¬(¬(¬x ⋅ ¬y) ⋅ ¬z))

b) x + y ̅(x ̅ + z): Using ⋅: x + y’⋅(x’ + z) Using −: (¬(¬(¬(¬(¬x’) ⋅ ¬y’) ⋅ ¬(¬z))))

c) (x + y ̅ ) ̅: Using ⋅: ¬(x + y’) Using −: (¬((¬(¬(¬(¬(¬(¬x’)) ⋅ ¬y’))))))

d) x ̅(x + y ̅ + z ̅): Using ⋅: x’⋅(x + y’ + z’) Using −: (¬(¬(¬((¬(¬((¬(¬(¬(¬(¬(¬(¬(¬(¬(¬(¬(¬x)))))))))))))))))

Part 2: Sets
Subset determination:
a) The set of people who speak English is a subset of the set of people who speak English with an Australian accent. b) The set of citrus fruits is a subset of the set of fruits. c) Neither set is a subset of the other as they represent different areas of study. d) Neither set is a subset of the other as they represent different groups of students.

True or False:
a) True b) True c) False d) True e) False f) True g) True h) True

Conclusions regarding set equality:
a) A = B cannot be concluded from A ∪ C = B ∪ C. b) A = B can be concluded from A ∩ C = B ∩ C. c) A = B can be concluded from both A ∪ C = B ∪ C and A ∩ C = B ∩ C.

Sets in Python:
a)

setA = {n for n in range(1000) if n % 100 % 43 == 0}

b)

setB = {n for n in range(1000) if n // 100 % 24 == 0}

c)

setC = setA.union(setB)

d)

setD = setA.intersection(setB)

e)

len_setA = len(setA)
len_setB = len(setB)
len_setC = len(setC)
len_setD = len(setD)

sum_AB = len_setA + len_setB
sum_CD = len_setC + len_setD

sum_AB == sum_CD # Returns True if the equation holds

f)

def cartesian(set1, set2):
return {(a, b) for a in set1 for b in set2}

set_product = cartesian(setD, {‘red’, ‘yellow’, ‘green’})

g)

traffic_light_colors = {‘red’, ‘yellow’, ‘green’}
cartesian_product = cartesian(setD, traffic_light_colors)

 

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