1. 01/12 Examples M=({p,q,r,s},{a,b},,p,{q,r}) a b p q r p q q q s
r s r r s
s s s
• *(p,ab)=(*(p,a),b)
=((*(p,),a),b)=((p,a),b)
=(q,b)=sF. Thus abL(M).
• (p,ab) (q,b) (s,).
• *(p,aa)=qF. Thus aaL(M).
• (p,aaa) (q,aa) (q,a) (q,).
• L(M)=a+b+
={an |n1}{bn |n1}. a a b b a b
a,b
•  L 

2. FAs with Partial s.t.f. a b p q r p q q q r r r Is this an FA?  No!
But …  This is an FA with partial state transition function, where
: QQ’Q
is a partial function.
THEOREM L(FA)=L(FA with part. s.t.f.). a a b b

3. Partial Functions and Total Functions
If f: XY is a function (mapping) or total function, then f(x) is defined for any x in X.
If f: XY is a partial function (mapping), then f(x) may be undefined for some x in X.
Ex.
Define f: NN by f(x)=y s.t. x=y2.
Then
f(1)=1, f(4)=2, f(9)=3, ….
But f(2), f(3), f(5), … are not defined.

4. (Reflexive and) Transitive Closure of a Binary Relation
R,SAA: binary relations on A
The composite of R and S:
SR = {(a,c) | (a,b) in R, (b,c) in S, for some c in A}.
n-th power of R
R0 = {(a,a) | a in A},
Rn+1 = RnR if n>0.
The transitive closure of R:
R+ = R1R2R3…
The reflexive transitive closure of R:
R* = R0R1R2…
•  L 

5. Reflexive and Transitive Closures (2)
A recursive definition of R*:
(a,a)R* for any aA.
(a,b)R* & (b,c)R(a,c)R*
A recursive definition of R+:
(a,b)R(a,b)R+
(a,b)R+ & (b,c)R(a,c)R+
THEOREM
(i) (a,b)R+ (a,b1)R, (b1,b2)R, …, (bk-1,bk)R, (bk,b)R for some b1,…,bk.
(ii) (a,b)R* Either a=b or (a,b)R+.

6. Examples of R* and R+ R={(a,b), (a,c), (b,c), (c,a)}
R0={(a,a),(b,b),(c,c)}
R1=R,
R2={(a,c),(a,a),(b,a),(c,b)}
R3={(a,a),(a,b),(a,c),(b,b),
(b,c),(c,c)}=R0R1
R*=R+={a,b,c}{a,b,c}
Consider the digraph represenation of binary relations.
For integers n and m, define nRmm=n+1. Then
nRkmm=n+k,
nR+mn<m, and
nR*mnm.