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Revert "notes"
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@Zune-Ahmed
Zune-Ahmed authored Sep 25, 2024
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32 changes: 17 additions & 15 deletions source/boolean-algebra/ch-boolean-algebra.ptx
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<?xml version="1.0" encoding="UTF-8"?>

<chapter xml:id="ch-boolean-algebra" xmlns:xi="http://www.w3.org/2001/XInclude">
<title>Boolean Algebra</title>

<introduction>
<p>
This chapter completes the preceding one by explaining how to ask Sage to decide whether a given lattice is a Boolean algebra.
We also illustrate basic operations with Boolean functions.
</p>
</introduction>
<!-- include sections -->
<xi:include href="sec-boolean-algebra.ptx" />
<xi:include href="sec-boolean-functions.ptx" />
</chapter>
<?xml version="1.0" encoding="UTF-8"?>

<chapter xml:id="ch-boolean-algebra" xmlns:xi="http://www.w3.org/2001/XInclude">
<title>Boolean Algebra</title>

<introduction>
<p>
This chapter completes the preceding one by explaining how to ask Sage to decide whether a given lattice is a Boolean algebra. We also illustrate basic operations with Boolean functions.
</p>
</introduction>

<!-- include sections -->
<xi:include href="sec-boolean-algebra.ptx" />
<xi:include href="sec-boolean-functions.ptx" />


</chapter>
194 changes: 98 additions & 96 deletions source/boolean-algebra/sec-boolean-algebra.ptx
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<section xml:id="sec-boolean-algebra" xmlns:xi="http://www.w3.org/2001/XInclude">
<title>Boolean Algebra</title><idx><h>Boolean Algebra</h></idx>
<aside>
<p>
In digital electronics, the principles of Boolean algebra guide the design of circuits such as multiplexers, demultiplexers, encoders, and decoders, essential components in telecommunications and signal processing.
</p>
</aside>
<p>
A Boolean algebra is a bounded lattice that is both complemented and distributive. Let's define the <c>is_boolean_algebra()</c> function to determine whether a given poset is a Boolean algebra. The function accepts a finite partially ordered set as input and returns a tuple containing a boolean value and a message explaining the result. Run the following cell to define the function and call it in other cells.
</p>
<sage>
<input>
def is_boolean_algebra(P):
try:
L = LatticePoset(P)
except ValueError as e:
return False, str(e)
if not L.is_bounded():
return False, "The lattice is not bounded."
if not L.is_distributive():
return False, "The lattice is not distributive."
if not L.is_complemented():
return False, "The lattice is not complemented."
return True, "The poset is a Boolean algebra."
</input>
</sage>
<p>
Let's check if the following poset is a Boolean algebra.
</p>
<sage>
<input>
S = Set([1, 2, 3, 4, 5, 6])

P = Poset((S, attrcall("divides")))

show(P)
</input>
</sage>
<sage>
<input>
is_boolean_algebra(P)
</input>
</sage>
<p>
When we pass <c>P</c> to the <c>is_boolean_algebra()</c> function, <c>LatticePoset()</c> raises an error because <c>P</c> is not a lattice. The <c>ValueError</c> provides more information about the absence of a top element. Therefore, <c>P</c> is not a Boolean algebra.
</p>
<sage>
<input>
T = Subsets(['a', 'b', 'c'])

Q = Poset((T, lambda x, y: x.issubset(y)))

Q.plot(vertex_size=3500, border=True)
</input>
</sage>
<sage>
<input>
is_boolean_algebra(Q)
</input>
</sage>
<p>
Let's examine the divisor lattice of 30:
</p>
<sage>
<input>
dl30 = Posets.DivisorLattice(30)
show(dl30)
is_boolean_algebra(dl30)
</input>
</sage>
<p>
Now for the divisor lattice of 20:
</p>
<sage>
<input>
dl20 = Posets.DivisorLattice(20)
show(dl20)
is_boolean_algebra(dl20)
</input>
</sage>
<p>
Here is a classic example in the field of computer science:
</p>
<sage>
<input>
B = posets.BooleanLattice(1)
show(B)
is_boolean_algebra(B)
</input>
</sage>
<aside>
<p>
Software engineers use Boolean algebra to simplify conditions, refactor code, and optimize logic. These skills are crucial for identifying bottlenecks and improving efficiency in real-time systems and performance-critical applications.
</p>
</aside>
</section>
<section xml:id="sec-boolean-algebra" xmlns:xi="http://www.w3.org/2001/XInclude">
<title>Boolean Algebra</title><idx><h>Boolean Algebra</h></idx>
<aside>
<p>
In digital electronics, the principles of Boolean algebra guide the design of circuits such as multiplexers, demultiplexers, encoders, and decoders, essential components in telecommunications and signal processing.
</p>
</aside>
<p>
A Boolean algebra is a bounded lattice that is both complemented and distributive. Let's define the <c>is_boolean_algebra()</c> function to determine whether a given poset is a Boolean algebra. The function accepts a finite partially ordered set as input and returns a tuple containing a boolean value and a message explaining the result. Run the following cell to define the function and call it in other cells.
</p>
<sage>
<input>
def is_boolean_algebra(P):
try:
L = LatticePoset(P)
except ValueError as e:
return False, str(e)
if not L.is_bounded():
return False, "The lattice is not bounded."
if not L.is_distributive():
return False, "The lattice is not distributive."
if not L.is_complemented():
return False, "The lattice is not complemented."
return True, "The poset is a Boolean algebra."
</input>
</sage>
<p>
Let's check if the following poset is a Boolean algebra.
</p>
<sage>
<input>
P = [1, 2, 3, 4, 5, 6]

divisibility_poset = Poset((P, attrcall("divides")))

show(divisibility_poset)
</input>
</sage>
<sage>
<input>
is_boolean_algebra(divisibility_poset)
</input>
</sage>
<p>
When we pass <m>P</m> to the <c>is_boolean_algebra()</c> function, <c>LatticePoset()</c> raises an error because <m>P</m> is not a lattice. The <c>ValueError</c> provides more information about the absence of a top element. Therefore, <c>divisibility_poset</c> is not a Boolean algebra.
</p>
<sage>
<input>
S = {'a', 'b', 'c'}

power_set = Subsets(S)

P = Poset((power_set, lambda x, y: x.issubset(y)))

P.plot(vertex_size=1000,vertex_shape="")
</input>
</sage>
<sage>
<input>
is_boolean_algebra(P)
</input>
</sage>
<p>
Let's examine the divisor lattice of 30:
</p>
<sage>
<input>
dl30 = Posets.DivisorLattice(30)
show(dl30)
is_boolean_algebra(dl30)
</input>
</sage>
<p>
Now for the divisor lattice of 20:
</p>
<sage>
<input>
dl20 = Posets.DivisorLattice(20)
show(dl20)
is_boolean_algebra(dl20)
</input>
</sage>
<p>
Here is a classic example in the field of computer science:
</p>
<sage>
<input>
B = posets.BooleanLattice(1)
show(B)
is_boolean_algebra(B)
</input>
</sage>
<aside>
<p>
Software engineers use Boolean algebra to simplify conditions, refactor code, and optimize logic. These skills are crucial for identifying bottlenecks and improving efficiency in real-time systems and performance-critical applications.
</p>
</aside>
</section>
122 changes: 61 additions & 61 deletions source/boolean-algebra/sec-boolean-definition.ptx
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<section xml:id="sec-boolean-definition" xmlns:xi="http://www.w3.org/2001/XInclude">
<title>Definition</title>
<p>
Boolean algebra is a branch of algebra that deals with variables that can take two values: true or false. This form of algebra is foundational in fields such as digital electronics and computer science where binary systems are prevalent.
</p>
<p>
In SageMath, Boolean algebra can be manipulated through various built-in functions designed for handling Boolean variables and expressions.
</p>
<aside>
<title>Notes</title>
<p>
Boolean algebra is crucial in digital circuit design. For example, simplifying the expression of a digital circuit can minimize the number of gates used, which reduces cost and power consumption. Techniques such as Karnaugh maps and Boolean simplification are common in the industry.
</p>
</aside>
<sage>
<input>
B = BooleanPolynomialRing(3, 'x')
f = B('x0 + x1 * x2')
show(f)
</input>
<output>
# Outputs the Boolean expression in polynomial form
</output>
</sage>
<p>
A common structure analyzed in Boolean algebra is the lattice. Below, we show how to work with a divisor lattice, which is a specific type of lattice useful in various mathematical computations.
</p>
<sage>
<input>
p = Posets.DivisorLattice(20)
p.show()
</input>
<output>
# This will display the lattice structure for divisors of the number 20.
</output>
</sage>
<p>
The following Sage cell checks whether the divisor lattice is distributive. A distributive lattice is one where the operations of join and meet distribute over each other.
</p>
<sage>
<input>
p = Posets.DivisorLattice(20)
p.is_distributive(certificate='True')
</input>
<output>
# Returns True if the lattice is distributive, along with a certificate if applicable.
</output>
</sage>
<p>
Another important property in Boolean algebra is the complemented lattice. A complemented lattice is one where every element has a complement in the lattice.
</p>
<sage>
<input>
p = Posets.DivisorLattice(20)
p.is_complemented(certificate='True')
</input>
<output>
# Returns True if the lattice is complemented, along with a certificate if applicable.
</output>
</sage>
</section>
<section xml:id="sec-boolean-definition" xmlns:xi="http://www.w3.org/2001/XInclude">
<title>Definition</title>
<p>
Boolean algebra is a branch of algebra that deals with variables that can take two values: true or false. This form of algebra is foundational in fields such as digital electronics and computer science where binary systems are prevalent.
</p>
<p>
In SageMath, Boolean algebra can be manipulated through various built-in functions designed for handling Boolean variables and expressions.
</p>
<aside>
<title>Notes</title>
<p>
Boolean algebra is crucial in digital circuit design. For example, simplifying the expression of a digital circuit can minimize the number of gates used, which reduces cost and power consumption. Techniques such as Karnaugh maps and Boolean simplification are common in the industry.
</p>
</aside>
<sage>
<input>
B = BooleanPolynomialRing(3, 'x')
f = B('x0 + x1 * x2')
show(f)
</input>
<output>
# Outputs the Boolean expression in polynomial form
</output>
</sage>
<p>
A common structure analyzed in Boolean algebra is the lattice. Below, we show how to work with a divisor lattice, which is a specific type of lattice useful in various mathematical computations.
</p>
<sage>
<input>
p = Posets.DivisorLattice(20)
p.show()
</input>
<output>
# This will display the lattice structure for divisors of the number 20.
</output>
</sage>
<p>
The following Sage cell checks whether the divisor lattice is distributive. A distributive lattice is one where the operations of join and meet distribute over each other.
</p>
<sage>
<input>
p = Posets.DivisorLattice(20)
p.is_distributive(certificate='True')
</input>
<output>
# Returns True if the lattice is distributive, along with a certificate if applicable.
</output>
</sage>
<p>
Another important property in Boolean algebra is the complemented lattice. A complemented lattice is one where every element has a complement in the lattice.
</p>
<sage>
<input>
p = Posets.DivisorLattice(20)
p.is_complemented(certificate='True')
</input>
<output>
# Returns True if the lattice is complemented, along with a certificate if applicable.
</output>
</sage>
</section>
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