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1. Propositional Calculus:
Boolean Algebra and
Simplification
CS 270: Mathematical Foundations
of Computer Science
Jeremy Johnson

2. Propositional Calculus
Topics
Motivation: Simplifying Conditional
Expressions
Rules of Boolean Algebra
Proofs Using Truth Tables
Simplification and Equational Reasoning
Logic Circuits
Functional Completeness of nand
Tautologies, Satisfiability and Automatic
Verification

3. 3
Programming Example
 Boolean expressions arise in conditional statements. It is
possible to abstract the relations with boolean variables
(propositions that are either true or false). Using this
abstraction one can reason and simplify conditional
statements.
 if ((a < b) || ((a >= b) && (c == d))) then { … } else { … }
 Let p denote the relation (a<b) and q denote the relation
(c == d). The above expression is then equal to
p || !p && q

4. 4
Programming Example (cont)
 The previous expression is equivalent (two expressions are
equivalent if they are true for the same values of the
variables occurring in the expressions) to a simpler
expression
 (p || !p && q) ≡ p || q
 We can see this since if p is true both expressions are true,
and if p is false, then !p is true and (!p && q) is true
exactly when q is true.

5. Boolean Algebra
 The Boolean operators ∨ and ∧ are
analogous to addition and multiplication
with true and false playing the roles of 1
and 0. Complement is used for negation.
 This provides a compact notation and
suggests appropriate algebraic
simplification
 Similar properties hold such as the
associative, commutative, and distributive
identities.

6. 6
Disjunctive Normal Form
 A Boolean expression is a Boolean function
 Any Boolean function can be written as a Boolean
expression
 Write a Boolean expression that evaluates to true for
each row in the truth table that is true and false for
other rows.
 The Boolean expression for a given row is the
conjunction of the variables that are true and the
negation of variables that are false.
 Take the disjunction of all such rows.
 E.G. (multiplexor function)
s x0 x1 f
0 0 0 0
0 0 1 0
0 1 0 1
0 1 1 1
1 0 0 0
1 0 1 1
1 1 0 0
1 1 1 1
(¬s ∧ x0 ∧ ¬x1 ) ∨ (¬s ∧ x0 ∧ x1 ) ∨ (s ∧ ¬x0 ∧ x1 ) ∨ (s ∧ x0 ∧ x1 )

7. 7
Sums of Products
 Disjunctive normal form, using the notation of
Boolean Algebra, corresponds to a sum of
products
 E.G. (multiplexor function)
s x0 x1 f
0 0 0 0
0 0 1 0
0 1 0 1
0 1 1 1
1 0 0 0
1 0 1 1
1 1 0 0
1 1 1 1

8. Properties of Boolean Algebra
 Boolean expressions can be simplified using rules of Boolean
algebra
 Identity law: A + 0 = A and A ● 1 = A.
 Zero and One laws (annhilator): A + 1 = 1 and A ● 0 = 0
 Inverse laws (complementation):
 Idempotent laws: A + A = A = A ● A
 Commutative laws: A + B = B + A and A ● B = B ● A.
 Associative laws: A + (B + C) = (A + B) + C and A ● (B ● C) = (A ● B) ● C.
 Distributive laws: A ● (B + C) = (A ● B) + (A ● C) and
A + (B ● C) = (A + B) ● (A + C)
 Absorption laws: A●(A+B) = A = A + A●B
 Double Negation: ̅̅
𝐴𝐴 = 𝐴𝐴
 DeMorgan’s laws:

9. Proof using Truth Tables
E.G. DeMorgan’s Law
A B (A ∨ B) ¬(A ∨ B) ¬A ∧ ¬B
0 0 0 1 1
0 1 1 0 0
1 0 1 0 0
1 1 1 0 0

10. Completeness
 Any identity from Boolean Algebra can be
proved using the properties listed
 A sufficient subset
 Associativity, commutativity, absorption, one
of the distributive laws and the two
complement laws

11. Proof of Double Negation
̅̅
𝑥𝑥 = ̅̅
𝑥𝑥 + 0
= ̅̅
𝑥𝑥 + 𝑥𝑥 ̅
𝑥𝑥
= ( ̅̅
𝑥𝑥 + 𝑥𝑥)( ̅̅
𝑥𝑥 + ̅
𝑥𝑥)
= ̅̅
𝑥𝑥 + 𝑥𝑥 � 1
= ( ̅̅
𝑥𝑥 + 𝑥𝑥)(𝑥𝑥 + ̅
𝑥𝑥)
= (𝑥𝑥 + ̅̅
𝑥𝑥)(𝑥𝑥 + ̅
𝑥𝑥)
= 𝑥𝑥 + ̅̅
𝑥𝑥 ̅
𝑥𝑥
= 𝑥𝑥 + 0
= 𝑥𝑥

12. 12
Simplification of Boolean
Expressions
 Simplifying multiplexor expression using Boolean algebra
 Equational reasoning: replace subexpressions by equivalent
expressions
 Reflexive, Symmetric, Transitive
 Verify that the boolean function corresponding to this
expression has the same truth table as the original function.

13. 13
Polynomial Algebra
Boolean operations via polynomial algebra
Assume x2 = x, y2 = y,…
Arithmetic performed mod 2
1⋅0 = 0⋅1 = 0⋅0 = 0, 1⋅1 = 1 [behaves like and]
0+0 = 1+1 = 0, 0+1 = 1+0 = 1 [behaves like xor]
And. x ∧ y via multiplication xy
Xor. x ⊕ y via addition x + y
Not. ¬x via 1+x
Or. x ∨ y via x+y+xy

14. 14
Polynomial Identities
Inverse Laws: (x ∧ ¬x) = F, (x ∨ ¬x) = T
x(1+x) = x + x2 = 2x = 0
x + (1+x) + x(1+x) = 1
Distributive Law: x ∧ (y ∨ z)
x(y + z + yz) = xy + xz + xyz = xy + xz + x2yz = xy + xz
+ xyxz
DeMorgan’ Laws: ¬x ∧ ¬y = ¬(x ∨ y) and
¬x ∨ ¬y = ¬(x ∧ y)
(1+x)(1+y) = 1 + x + y + xy
(1+x)+(1+y) + (1+x)(1+y) = (1+x) + (1+y) + (1 + x + y
+ xy) = 1+xy

15. 15
Logic Circuits
 A single line labeled x is a logic circuit. One end is the input
and the other is the output. If A and B are logic circuits so
are:
 and gate
 or gate
 inverter (not)
A
B
A
A
B

16. XOR
“One or the other, but
not both”
Notation for circuits:
x
y
x y x ⊕ y
0 0 0
0 1 1
1 0 1
1 1 0
x
y

17. 17
Multi-Input Gates
 Cascaded and gate
 A ∧ B ∧ C ∧ D ≡ ((A ∧ B) ∧ C) ∧ D ≡ (A ∧ B) ∧ (C ∧ D) ≡ A ∧ (B ∧ (C ∧ D))
A
B
C
D
A
B
C
D
A
B
C
D

18. 18
Logic Circuits
Given a boolean expression it is easy to write
down the corresponding logic circuit
Here is the circuit for the original multiplexor
expression
x0
x1
s

19. 19
Logic Circuits
Here is the circuit for the simplified
multiplexor expression
x0
x1
s

20. 20
Nand
Nand – negation of the conjunction
operation:
A nand gate is an inverted and gate:
x y x | y
0 0 1
0 1 1
1 0 1
1 1 0

21. 21
Implementing Logic Gates with
Transistors
output
gate
+V
ground
A Transistor NOT Gate
A NAND B
A
+V
ground
A Transistor NAND Gate
B

22. Nand is functionally complete
Using DeMorgan’s Law, ¬ and ∧ are enough to
generate all Boolean functions
All boolean functions can be implemented
using nand gates (and, or and not can be
implemented using nand)
not:
and:
or:
x y x | y
0 0 1
0 1 1
1 0 1
1 1 0

23. 23
Arithmetic via Logic
Full Adder Used to add to binary numbers stored
as an array of bits using carry ripple addition
Three binary inputs
a, b and CarryIn
Two binary outputs
Sum and CarryOut such that a + b + CarryIn =
2*CarryOut + Sum
Sum
CarryIn
CarryOut
a
b
Carry 110
A 101
B 111
A+B = 1100

24. 24
Logic Circuit Lab Exercise
Derive a truth table for the output bits (Sum and
CarryOut) of a full adder.
Using the truth table derive a sum of products
expression for Sum and CarryOut. Draw a circuit
for these expressions.
Using properties of Boolean algebra simplify your
expressions. Draw the simplified circuits.
Sum
CarryIn
CarryOut
a
b

25. 25
Tautologies
A tautology is a boolean expression that is always
true, independent of the values of the variables
occurring in the expression. The properties of
Boolean Algebra are examples of tautologies.
Tautologies can be verified using truth tables. The
truth table below shows that x → y ≡ ¬ x ∨ y
x y x → y ¬ x ∨ y
0 0 1 1
0 1 1 1
1 0 0 0
1 1 1 1

26. 26
Exercise
Derive the tautology
x → y ≡ ¬ x ∨ y
from the sum of products expression
obtained from the truth table for x → y.
You will need to use properties of Boolean
algebra to simplify the sum of products
expression to obtain the desired
equivalence.

27. 27
Solution
Derive the tautology
x → y ≡ ¬ x ∨ y
𝑥𝑥 → 𝑦𝑦 ≡ (¬𝑥𝑥 ∧ ¬𝑦𝑦) ∨ ¬𝑥𝑥 ∧ 𝑦𝑦 ∨ 𝑥𝑥 ∧ 𝑦𝑦
≡ (¬𝑥𝑥 ∧ ¬𝑦𝑦) ∨ (¬𝑥𝑥 ∧ 𝑦𝑦) ∨ ¬𝑥𝑥 ∧ 𝑦𝑦 ∨ 𝑥𝑥 ∧ 𝑦𝑦
≡ ¬𝑥𝑥 ∧ (¬𝑦𝑦 ∨ 𝑦𝑦) ∨ ¬𝑥𝑥 ∨ 𝑥𝑥 ∧ 𝑦𝑦
≡ ¬𝑥𝑥 ∧ #𝑡𝑡 ∨ (#𝑡𝑡 ∧ 𝑦𝑦)
≡ ¬𝑥𝑥 ∨ 𝑦𝑦
x y x → y
0 0 1
0 1 1
1 0 0
1 1 1

28. 28
Tautology Checker
 A program can be written to check to see if a Boolean
expression is a tautology.
 Simply generate all possible truth assignments for the
variables occurring in the expression and evaluate the
expression with its variables set to each of these assignments.
If the evaluated expressions are always true, then the given
Boolean expression is a tautology.
 Two Boolean expressions E1 and E2 are equivalent if E1 ≡ E2 is
a tautology.

29. Proof using Truth Tables
E.G. DeMorgan’s Law
A B (A ∨ B) ¬(A ∨ B) ¬A ∧ ¬B
0 0 0 1 1
0 1 1 0 0
1 0 1 0 0
1 1 1 0 0

30. Satisfiability
 A formula is satisfiable if there is an assignment
to the variables that make it true
 Checking to see if a formula f is satisfiable can be
done by searching a truth table for a true entry
 Exponential in the number of variables
 Does not appear to be a polynomial time algorithm
(satisfiability is NP-complete)
 There are efficient satisfiability checkers that work
well on many practical problems
 f is a tautology ¬ f is not satisfiable.

31. Arguments
 An argument is a list of premises followed by a
conclusion
 ϕ1,…,ϕn ∴ ψ
 An argument is valid if any assignment that
makes the premises true also makes the
conclusion true
 An argument is invalid if there is some
assignment, counter example, that makes the
premises true but does not make the conclusion
true
 Equivalent to ϕ1 ∧ … ∧ ϕn → ψ

32. 32
Lab Exercise
Derive a truth table for the output bits (Sum and
CarryOut) of a full adder.
Using the truth table derive a sum of products
expression for Sum and CarryOut. Draw a circuit
for these expressions.
Using properties of Boolean algebra simplify your
expressions. Draw the simplified circuits.
Sum
CarryIn
CarryOut
a
b