Download DecModelingBooleanAlgebraSpreadSheets FAS2015 3/20

Decision Modeling Using
Boolean Algebra in
Spreadsheets
Bryan Barney Reagan, Ph.D.
Saint Leo University
Law of Excluded Middle
Aristotle (384 – 322 BC)
“For every assertion must, as is admitted, be
either true or false, ….”
- Aristotle, Section 1, Part 4, “Categories”, The
Organon, Translated by E. M. Edghill (2011)
Page 15.
Bust of Aristotle. Marble, Roman
copy after a Greek bronze original
By Lysippos from 330 BC; Photo
http://commons.wikimedia.org/wiki/Lysippos
Roots of X*X = X
George Boole (1815 -1864 A.D.)
Founded the field of Boolean Algebra in his two
works: Mathematical Analysis of Logic (1847)
and An Investigation of the Laws of Thought
(1854)
http://www.maa.org
In order to use simple mathematical operations
to model logic. In order for X*X to equal X, he
used the values zero (0) and one (1) for false
and true. (Boole, George, Investigation of the
Laws of Thought, Chapter 3. pp. 39-51)
Boole’s Notation
Negation
𝑩 = 𝟎, 𝟏
(∀ 𝐛) 𝐛 ∈ 𝑩 → (∃ 𝟏 − 𝒃 ∈ 𝑩)
((𝟏 − 𝒃) ≠ 𝒃)
Conjunction
∀𝐛 𝐛∈𝑩 → 𝒃∗𝟏=𝒃
𝒃∗𝟎=𝟎
∀ 𝐚, 𝐛 𝐚, 𝐛 ∈ 𝑩 → 𝒂 ∗ 𝒃 = 𝒃 ∗ 𝒂
Disjunction
Disjunction is denoted with +, but it is not simple addition since it must
not produce results greater than 1.
We can derive an expression for “A + B” using DeMorgan’s Thereom,
with which Boole was familiar.
A + B = 1 – ((1-A)*(1-B))
= 1 – (1 – A – B + AB)
= 1 – 1 + A + B – AB
= A + B - AB
Disjunction
∀𝐛 𝐛∈𝑩 → 𝒃+𝟏=𝟏
𝒃+𝟎=𝒃
∀ 𝐚, 𝐛 𝐚, 𝐛 ∈ 𝑩 → 𝒂 + 𝒃 = 𝒃 + 𝒂
Distributive Properties in Boolean Algebras
∀ 𝐚, 𝐛, 𝐜 𝐚, 𝒃, 𝒄 ∈ 𝑩 → 𝒂 ∗ 𝒃 + 𝒄) = (𝒂 ∗ 𝒃) + (𝒂 ∗ 𝒄
∀ 𝐚, 𝐛, 𝐜 𝐚, 𝒃, 𝒄 ∈ 𝑩 → 𝒂 + 𝒃 ∗ 𝒄) = (𝒂 + 𝒃) ∗ (𝒂 + 𝒄
Logical Values in Spreadsheets
Modern spreadsheets such as Excel support the Logical Values TRUE and FALSE ,
which may be entered as text or the result of comparisons using relational
operators.
Logical /Boolean Operators in Excel
Operator
Excel Notation
Negation
=NOT(B12)
Conjunction =AND(B12, C12)
=AND(B12:C12)
Disjunction =OR(B12, C12)
=OR(B12:C12)
Conditional Formulas in Excel
=IF( Test Expression, Value if True, Value if False)
=IF(B12>500, 400, 0)
=IF(AND(B12>500,C12>0), 400, 0)
=IF(B12=MAX($B$2:$B$26),”*”,””)
Conversion Between Logical and Binary Values
• To convert from Logical to Binary, Excel allows multiplication of the
values {False, True} by 1 to produce {0, 1}.
= B12 * 1
• To convert from Binary values to Logical, you can use the OR function
with a single parameter to convert an value from {0, 1} to {False,
True}.
= OR(C12)
Solver Constraints
Thank you
[email protected]