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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?

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.

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

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

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 fg 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)2f(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

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T

F

F

F

Truth table for fExample (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:
- (pq r) (pqr) (pqr) (pqr) (pqr) (pqr)

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:

pp nand p

and

p q (p nand q) nand (p nand 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)

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