EE1J2 - Slide 1

EE1J2 – Discrete Maths Lecture 5 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)

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Truth tables So far we have seen how to build a truth

table T for a given formula f in propositional logic

Today we’ll look at the opposite problem: Given a set of atomic propositions p1,…,pN and a truth table T, can we construct a formula f such that T is the truth table for f ?

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Adequacy A set of propositional connectives is

adequate if For any set of atomic propositions p1,

…,pN and For any truth table for these propositions, There is a formula involving only the

given connectives, which has the given truth table.

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Adequacy 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

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Some more definitions… f1, f2,…,fn a set of n formulae

f1 f2 … fn is called the disjunction of f1,

f2,…,fn

f1 f2 … fn is 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

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Disjunctive Normal Form A formula is in Disjunctive Normal Form

(DNF) if it is a disjunction of conjunctions of literals.

Examples: (p, q, r and s atomic propositions)p q

(p) (q)

(p q) (p r s)

…

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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)….

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Truth Functions A truth function is 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}.

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

p q  

T T F

T F T

F T T

F F F

22 rows

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Truth functions

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

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

23 rows

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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

Corollary: Any formula is logically equivalent to a formula in disjunctive normal form

Corollary: {, ,} is an adequate set of connectives

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Proof of theorem Let p1, 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

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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

fr takes the truth value T for the rth row of the truth table and F for all other rows.

by rowin F value theassigned is if

by rowin T value theassigned is if )(

rpp

rpprf

ii

iii

irf

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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

Rrrr fff ...21

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Corollary 1 Any formula is logically equivalent to a

formula in disjunctive normal form Any formula g defines a truth table By the above theorem there is a formula f in

disjunctive normal form which has the same truth table as g

Hence f is logically equivalent to g

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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 .

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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 g is logically equivalent to (f g)

f g is logically equivalent to f g

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Corollary 4 {, } and {, } are both adequate sets of

connectives Proof – homework

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The symbol The symbol means logical equivalence Next look at some standard equivalences

using the set {, ,}

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Standard equivalences

(a) f g g f, f g g f Commutativity

(b) (f g) h f (g h)(f g) h f (g h)

Associativity

(c) f (g h) (f g) (f h)f (g h) (f g) (f h)

Distributivity

(d) (f g) (f) (g)(f g) (f) (g)

De Morgan’s Laws

(e) f f Rule of double negation

(f) f f is a tautologyf f is a contradiction

 

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Theorem 2

Let be a truth function. Then there is a formula in Conjunctive Normal Form (CNF) whose truth table is given by

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DNF - Example

Let p, q and 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 theorem.

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Truth table for f(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

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Example (continued) From row 1: (pq r) From row 2: (pq r) From row 3: (pqr) From row 4: (pqr) From row 5: (pqr) From row 7: (pqr) Hence the desired formula is:(pq r)(pq r)(pqr)(pqr)

(pqr)(pqr)

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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

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nand and nor gates Most common gates are nand gates and nor

gates. Their truth tables are given by Truth tables for nand and nor

p q p nand q p nor q

T T F F

T F T F

F T T F

F F T T

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Theorem 3Adequacy of nand and nor Theorem: The sets {nand} and {nor} are both

adequate Proof

{nand}: Since {, } is adequate, 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)

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

p p nor p

p q (p nor p) nor (q nor q)

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Summary of Lecture 5

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)