EE1J2 – Discrete Maths Lecture 6

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EE1J2 – Discrete Maths Lecture 6 - PowerPoint PPT Presentation

EE1J2 – Discrete Maths Lecture 6. Adequacy of a set of connectives Disjunctive and conjunctive normal form Adequacy of { , , , }, { , , }, { , } and { , } Every formula is logically equivalent to one in conjunctive normal form (or disjunctive normal form). Truth tables.

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EE1J2 – Discrete Maths Lecture 6
• Adequacy of a set of connectives
• Disjunctive and conjunctive normal form
• Adequacy of {, , , }, {, , }, {, } and {, }
• Every formula is logically equivalent to one in conjunctive normal form (or disjunctive normal form)
Truth tables
• So far we have seen how to build a truth table Tfor a given formula fin propositional logic
• Today we’ll look at the opposite problem: Given a set of atomic propositions p1,…,pNand a truth table T, can we construct a formula f such that T is the truth table for f?
• A set of propositional connectives is adequate if
• For any set of atomic propositions p1,…,pNand
• For any truth table for these propositions,
• There is a formula involving only the given connectives, which has the given truth table.
• The goal of today’s lecture is to show that the set {, , , } is adequate and contains redundancy, in the sense that it contains subsets which are themselves adequate
• We shall also introduce other sets of adequate connectives
Some more definitions…
• f1, f2,…,fn a set of n formulae
• f1 f2… fnis called the disjunction of f1, f2,…,fn
• f1 f2… fnis called the conjunction of f1, f2,…,fn
• Let p be an atomic proposition. A formula of the form p or p is called a literal
Disjunctive Normal Form
• A formula is in Disjunctive Normal Form(DNF) if it is a disjunction of conjunctions of literals.
• Examples: (p, q, rand s atomic propositions)

p  q

(p)  (q)

(p  q) (p  r  s)

Conjunctive Normal Form
• A formula is in Conjunctive Normal Form (CNF) if it is a conjunction of disjunctions of literals
• Examples:

p  q

(p)  (q)

(p  q)  (p)

….

Truth Functions
• A truth functionis a function  which assigns to a set of atomic propositions {p1,…,pN} a truth table (p1,…,pN) in which one of the truth values T or F is assigned to each possible assignment of truth values to the atomic propositions {p1,…,pN}.
p

q

T

T

F

T

F

T

F

T

T

F

F

F

Truth functions
• p, q and r atomic propositions
• Example truth function in {p, q}

22 rows

p

q

r

T

T

T

T

T

T

F

F

T

F

T

T

T

F

F

T

F

T

T

F

F

T

F

T

F

F

T

F

F

F

F

F

Truth functions
• Example truth function in 3 atomic propositions {p, q, r}

23 rows

First Theorem (Disjunctive Normal Form)
• Theorem: Let  be a truth function. Then there is a formula in disjunctive normal form whose truth table is given by 
The symbol 
• Recall that the symbol  means logical equivalence
• Two formulae are logically equivalent if they have the same truth table
Corollaries to First Theorem
• Corollary 1: Any formula is logically equivalent to a formula in disjunctive normal form
• To see this:
• Let f be a formula
• Construct the truth table for f
• By the First Theorem there is a formula g in DNF which has this truth table
• Then fg by definition
Corollaries to First Theorem
• Corollary 2: {, ,} is an adequate set of connectives
• To see this, note:
• Given any truth table we can construct a formula f in DNF which has that truth table
• By definition of DNF, f only involves the connectives , ,
Proof of First Theorem
• Letp1, p2,…,pn be the atomic propositions
• Want a formula  in disjunctive normal form whose truth table is given by 
• If  assigns the value F to every row of the truth table, just choose  = 
• Otherwise, there will be at least one row for which the truth value is T. Let that row be row r
Proof (continued)
• let be the formula defined by:
• Let fr be the conjunction

f(r)1 f(r)2f(r)3…f(r)n

• frtakes the truth value Tfor the rth row of the truth table and F for all other rows.
Proof (continued)
• Suppose that there are R rows r1,…,rR for which the truth value is T.
• Define =
• Clearly  is in disjunctive normal form
• By construction  has the truth table defined by 
DNF - Example

Let p, qand r be atomic propositions

Consider f = (p(q  r))  ((p  q)  r)

How do we put this in disjunctive normal form?

Use the construction from the proof of the First Theorem (DNF) from lecture 5.

(p

(q

r))

((p

q)

r)

T

T

T

T

T

T

T

T

T

T

T

T

F

T

F

F

T

T

T

T

F

F

T

T

F

T

T

T

T

F

F

T

T

T

T

F

T

F

T

T

F

F

T

F

F

T

T

T

T

T

F

T

T

T

T

F

T

T

F

F

F

F

T

T

F

F

F

T

F

T

T

T

F

T

F

T

T

F

T

F

T

F

F

F

T

F

F

F

Truth table for f
Example (continued)
• First identify the rows in the truth table for which f is true
• Then use these rows to construct the components of the DNF version of f:
• From row 1: (p  q  r)
• From row 2: (p  q  r)
• From row 3: (p  q  r)
• From row 4: (p  q  r)
• From row 5: (p  q  r)
• From row 7: (p  q  r)
Example (continued)
• Now combine these using ‘or’ symbols to obtain the desired formula in DNF:
• (pq r)  (pqr)  (pqr)  (pqr)  (pqr)  (pqr)
Corollary 1
• Any formula is logically equivalent to a formula in disjunctive normal form
• Any formula gdefines a truth table
• By the above theorem there is a formula f in disjunctive normal form which has the same truth table as g
• Hencef is logically equivalent to g
Corollary 2
• {, ,} is an adequate set of connectives
• From the theorem, any truth table can be satisfied by a formula in disjunctive normal form.
• By definition, such a formula only employs the connectives ,  and .
Corollary 3
• {, } is an adequate set of connectives
• Enough to show that  and  can both be expressed in terms of the symbols  and .
• To see this, note that if f and g are formulae in propositional logic:

f  gis logically equivalent to (f g)

f  g is logically equivalent to f  g

Corollary 4
• {, } and {, } are both adequate sets of connectives
• Proof – homework
Theorem 2
• Let  be a truth function. Then there is a formula in Conjunctive Normal Form (CNF) whose truth table is given by 
Switching Circuits
• Connections between propositional logic and switching circuits
• Can think of a truth table as indicating the ‘output’ of a particular circuit once its inputs have been set to ‘On’ or ‘Off’
• Now know that any desired behaviour can be obtained provided that the gates of the circuit can instantiate the connectives ,  and 
Truth tables for nand and nor

p

q

p nand q

pnorq

T

T

F

F

T

F

T

F

F

T

T

F

F

F

T

T

nand and nor gates
• Most common gates are nand gates and nor gates. Their truth tables are given by
Theorem 3Adequacy of nand and nor
• Theorem: The sets {nand} and {nor} are both adequate
• Proof

{nand}: Since {, } is adequate, it is enough to show that  and  can be expressed in terms of nand.

Let p and q be atomic propositions. Then:

pp nand p

and

p  q (p nand q) nand (p nand q)

Proof (continued)
• For {nor}: It is enough to notice that:

pp nor p

p  q (p nor p) nor (q nor q)

Summary of Lecture 6

Adequacy of a set of connectives defined

• Disjunctive and conjunctive normal form defined
• Adequacy of {, , , }, {, , }, {, }, {, }, {nand} and {nor}
• Every formula is logically equivalent to one in disjunctive normal form (DNF)