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)

2. 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 ?

3. 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.

4. 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

5. 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

6. 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) …

7. 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) ….

8. 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}.

9. Truth functions p, q and r atomic propositions
Example truth function in {p, q} p q T F
22 rows

10. Truth functions p q r T F
Example truth function in 3 atomic propositions {p, q, r}
23 rows

11. 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

12. 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

13. 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.

14. 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 

15. 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

16. 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 .

17. 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

18. Corollary 4 {, } and {, } are both adequate sets of connectives
Proof – homework

19. The symbol  The symbol  means logical equivalence
Next look at some standard equivalences using the set {, ,}

20. Standard equivalences
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 tautology
f  f is a contradiction

21. Theorem 2
Let  be a truth function. Then there is a formula in Conjunctive Normal Form (CNF) whose truth table is given by 

22. 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.

23. Truth table for f (p  (q
r))
((p q) r) T F

24. Example (continued) From row 1: (pq r) From row 2: (pq r)
Hence the desired formula is:
(pq r)(pq r)(pqr)(pqr) (pqr)(pqr)

25. 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 

26. 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 F

27. Theorem 3 Adequacy 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)

28. Proof (continued) For {nor}: It is enough to notice that: p  p nor p
p  q  (p nor p) nor (q nor q)

29. 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)