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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.

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Ee1j2 discrete maths lecture 6 l.jpg
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)


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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?


Adequacy l.jpg
Adequacy

  • 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.


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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… 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


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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, rand s atomic propositions)

    p  q

    (p)  (q)

    (p  q) (p  r  s)


Conjunctive normal form l.jpg
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 l.jpg
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}.


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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


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p

q

r

T

T

T

T

T

T

F

F

T

F

T

T

T

F

F

T

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T

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T

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T

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Truth functions

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

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 


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The symbol

  • Recall that the symbol  means logical equivalence

  • Two formulae are logically equivalent if they have the same truth table


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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


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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 l.jpg
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


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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

  • frtakes the truth value Tfor the rth row of the truth table and F for all other rows.


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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 


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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.


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(p

(q

r))

((p

q)

r)

T

T

T

T

T

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T

T

T

T

T

T

F

T

F

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F

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T

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F

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T

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T

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Truth table for f


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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)


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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)


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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


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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  gis 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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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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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 


Nand and nor gates l.jpg

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


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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)


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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 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)


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