1. Propositional Equivalence
Goal: Show how propositional equivalences are established & introduce the most important such equivalences.

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Equivalence
Name
p  T ≡ p; p  F ≡ p
Identity
p  T ≡ T; p  F ≡ F
Domination
p  p ≡ p; p  p ≡ p
Idempotent
( p) ≡ p
Double negation
p  q ≡ q  p; p  q ≡ q  p
Commutative
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Equivalence
Name
(p  q)  r ≡ p  (q  r )
(p  q)  r ≡ p  (q  r )
Associative
p  (q  r ) ≡ (p  q)  (p  r )
p  (q  r ) ≡ (p  q)  (p  r )
Distributive
(p  q) ≡  p   q
(p  q) ≡  p   q
DeMorgan
p  (p  q ) ≡ p
p  (p  q ) ≡ p
Absorption
p  p ≡ T
p  p ≡ F
Negation
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Exercise
A tautology is a compound proposition that always is true.
Is the following a tautology?
( (p  q)  ( p  r) )  (q  r)
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5. The Satisfiability Problem
Given a function of Boolean variables, is there an assignment of values to its variables that makes it true?
Is f( p, q, r ) = ( (p  q)  ( p  r) )  (q  r) satisfiable?
Satisfiability is important in CS theory, algorithms, program correctness, AI, hardware design, etc.
Algorithm
Construct the truth table.
If any assignment (row) evaluates to true, return true;
Else return false.
If the formula has n variables, how many rows does the truth table have?
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6. An example satisfiability problem
Let p( row, col, n ) denote the proposition
“Box( row, col ) contains number n.”
Using such propositions, design a compound proposition that is satisfiable if & only if
n appears in some box, for 1 ≤ n ≤ 4. 1 3 4 1
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Problem
Give logical expressions for a 2-bit adder, where
true corresponds to 1
false corresponds to 0
For example, = 100.
Input:
Operand 1: a1 a0
Operand 2: b1 b0
Output
s2 s1 s0
That is, define 3 Boolean functions:
s0( a1 , a0 , b1 , b0 ) = ?
s1( a1 , a0 , b1 , b0 ) = ?
s2( a1 , a0 , b1 , b0 ) = ?
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Can you define a Boolean function in the C programming language?
boolean[] adder( boolean a1, boolean a0, boolean b1, boolean b0 ) { }
Or, for an n-bit adder:
boolean[] adder( boolean[] a, boolean[] b ) { }
For an n-bit adder, it may be useful to compute, for 0 ≤ i ≤ n, a sum bit, si and a carry bit, ci.
For the sum bit, si, we may use:
si = ai  bi  ci-1, where c-1 = 0 and sn = cn-1.
The equation above is called a recurrence equation.
What is a recurrence equation for the carry bit, ci?
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Unraveling the for loop, suggests a diagram:
Each box above has 3 inputs & 2 outputs, and is called a full adder.
A harder problem: Compute these sum & carry bits in parallel. sn
sn-1 s2 s1 s0 c0
cn-2 c2 c1
cn-1
an-1
bn-1 a2 b2 a1 b1 a0 b0
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10. Copyright © Peter Cappello 2011
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Copyright © Peter Cappello 2011