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25 Nov 2002: But the map (a,b)-->(b,a) is easily seen to be a bijection between AxB and BxA, from which it follows that BxA has cardinality mn. ... If m=1 then the result is obvious, since the map (1,x)-->x is a bijection between AxB and B, B has cardinality n, and

24 Sep 2002: 1y wŽ. msyj s 1q j 2 j 3 j 4 jw q5w q5w q wq b b jÝ1 2 5j1y wŽ. ... P a, d s b 2 r q b b r q b 2 r , 5.8Ž.

20 May 2002: 15. p. 243, lines 6 and 9: replace "(., f((r - 1)b/N /- |B_r| (b/N)2))" by "(., f((r - 1)b/N) /- |B_r| (b/N)2)". ... the brackets are misplaced). 16. p. 247, line 6: delete "<= B b3 N(-2)" (I am not quite sure here.).

23 Sep 2002: Tu. 10 (One class, week 2). PROF. B. J. HEALEgoism, pleasure and value. ... 11 (Sixteen lectures). DR R. B. B. WARDY (CL)†Plato Republic 473–535.

1 Oct 2002: iff for every bundle b of S there exists a bundle b′ of S′ such that b = b′. ... b′ B | e b′ b′ b}. Proposition 6. For every b B and every e Eb the set deeb is a bundle in B.For every finite and compatible X B

31 Jan 2002: Searching for the optimum Root-Goal path. The methods of the previous section found some path between root and goal. A rather harder problem is to find the best path. For this we need to exploit the cost figures in the graph. British Museum. This

27 Aug 2002: xO:N"I(B B[CCFN[CjA@R[kND@5OQH5B"s ilkJFRO:BR[kCI(OQB[RDN[OQ5>5RDO:@H@]ºRDk5CR[@RFV ... Hc?5N[@5e"CBDBS@]NFRDCP¤ÌFHJIB[>?5?J@B[CSOQRR"FCB-R[OQXC=RD@Py-FVQFe"N[@B[B-R[kCuVFH5Ch¤ÇB[BD>5XOQH5¿R[kJFR-R[k5C?JCfI

26 Sep 2002: hom(A,f,B,g) {H A B. (f AA A) (g B B B) (xA. ... 1. [| J : A -> B; K : B -> C; f : A A -> A; g : B B -> B;.

11 Mar 2002: g : 〈aa〉 〈bb〉 Iisuch that g extends f (i.e. g f).Similarly, for every b B. ... For finite A and B,. (A, a) kω (B, b)if, and only if,.

13 Sep 2002: Negation Normal Form. 1. Get rid of and , leaving just , , :. A B ' (A B) (B A). A B ' A B. ... r). A B, B A(r). (A B) (B A)(r). III Logic and Proof 15.

25 Apr 2002: Next, we perhaps note that if b=a. 2. then we have the equation x. ... a-3t)y. 2. (b-2at3t. 2. y) c-btat. 2. t. 3. This gives us a'=a-3t and b'=b-2at3t.

27 Aug 2002: 2H0&<(?:/B?&, B>,B+:DY&H]r%3&$0XB?3&7+32H(&$0(:/B?&<:/(¤Ç 4r&<(?:/DA3., B+&,X03&$4ã98, 0152HDIH, 9?:, ;.&YL N KÇJOF4T:/B?IH,. ... 015t2H9TÚ wy (?&$B Ú9 f Æ ÚûK£ÇYO u Ç w »# È }%'&C7$, 0&Z=.2:/B)B?&<£,7"B)B+-, B>B+3&72H0-5:/B+:20-,.

26 Sep 2002: Next, use the other &E axiom:. [[A & B]] [[B]] [[B]] [[C]][[A & B]] [[C]]. ... A. [[(A ) ]] [[A]]) (. B. [[B (B )]]). Here the premise is a schematic rule.

11 Nov 2002: k (a, b) | b, which contradicts (a, b) being the highest common factor. ... Proof: () (a, b) | a and (a, b) | b, so (a, b) | (ax by) = c.

15 Mar 2002: a) Show that, if A k A′ and B k B′, then A B k A′ B′.(b) Show that, for n,m k, nA k mA. ... Suppose that for each i < n, (A,ai,ai1) 3p(B,bi,bi1). Show that (A, a) p (B, b) (i.e.

26 Sep 2002: Γ a AΓ,b B a A. The product introduction rule is handled like -introduction. ... Γ A type Γ, priΓ,b,B A b(priΓ,b,B) B(priΓ,b,B)Γ lambda(b) Π(A,B). In Isabelle, the Type Theory parser, read rule, is

18 Sep 2002: U4Z+[0:J81:<;1G/ '9Ja' m /HJKml'54p)L<;1G/ '9Jq 12GgK1 D 02'9Y54lW[=WX'(e5+"7(e"'(WY5)I+J8130;79e5<81:<Vb#b#b#bq /HJ8021T4rYWX1G/ '9JK0 D ... 13'Ô+JWX'(e5+"b% "b c@b A &(xs YK0=PpªV> /HJ[0GÄg ª O9?

26 Sep 2002: A B. ABa : A b : B〈a,b〉 : AB. 14 Chapter 2. ... n : Nat b : B(zero). [x : Nat ; y : B(x).

26 Sep 2002: A,B | B A,B | AA,B | B A :right. AB | B A :left. ... R(?a,f(?a)) and R(?b,? b). 1.4 Parameters in quantifier rules.

26 Sep 2002: α = [. , 0, 0, a, 0, b, 1 a b, 1, 1 a b, b, 0, a, 0, 0,. ] ... α(1) = 3[. , 0, 0, b 13,13 b, b, 1 2b, b, 1.

26 Sep 2002: i,j) evnodd A b = ((i,j) A (ij) mod 2 = b)evnodd (A B) b = evnodd A b evnodd B b. ... goalw thy [evnodd_def] "evnodd (A - B) b = evnodd A b - evnodd B b";by (Blast_tac 1);qed "evnodd_Diff";.

26 Sep 2002: parts (sees Spy evs) B=B’ P=P’. Although it is not used in later proofs, unicity of nonces is important. ... Hash {|Key(shrK A), Agent A, B, NA, P|}. parts (sees Spy evs) B=B’ P=P’.

13 Sep 2002: Step 1. Eliminate and by repeatedly applying the following equivalences:. A B ' ( A B) (B A). ... A A A. B A B. A B B A. (B C) ( A B A C).

11 Nov 2002: Then either B B.x B or C C.x C. In both cases B B. ... Then B B. C C.x B C so B B.x B ) (c CC).

17 Dec 2002: 3 = # / " -# <= )B+ # 2" 84-"$/, % ;$/ <& 2" =! =! 4( "4 1B 8. B "" " B+ "0%'(" , 8 7 ,-G%@ 3 , 78 ,"G" , 8: ,BGA"9. , ... J /L J ,"8 ,L. ; B ( ) 7 >- B+ # & B # #! $/ (O ". %"<; 7'" B )% " 47 >- ,08 DB!

24 Sep 2002: j j, p s b b w j p q y3b 2 q10 b b y3b 2 w 2 j p 2Ž. ... G s b 2 q b 2 r q b b r ,Ž.

26 Sep 2002: a =α a. a =α b. b =α a. a =α b P (b). ... arithmetic is developed up to the theorem a mod b (a/b) b = a.Reflection works well in higher-order reasoning.

31 Oct 2002: n2. (b) Let X be any random variable which takes on values 0,1,2,. ... a) Determine the equilibrium points. (b) Determine the stability of each of the equilibrium points.

26 Sep 2002: variable; the printer chooses a unique name. A loose bound variable is printed B.n. ... op thenp: ’a parser ’b parser -> (’a’b)parser. Composition of functionals: parses one phrase, then another.

28 Mar 2002: X <Y6N12Z[2JOL2> &2 Q]2_ ]a bcO$$d2JOL2> &2 Qef<@;)A)B)B2B)AN C bghd&.ji# kl5& m@onqp$,7&h.$M srt< ; A)B)B)BA1CR ] ; ... $M. { rt t;A)B)B)B2AN OR ];2AB)B)B)AN]STM= <p$#,&. $KJ#),)h1&. $O 9 }S@' ] bcO$$#)dJ#),)h1&.

6 Mar 2002: Programs are clearer, as in this definition of app:. app(A-B, B-C, A-C). ... normalise(A, C) :-. flat(A, B-[]),. B = [T | L],. build(L, T, C).

12 Sep 2002: B, consisting of a reasonable proportion of the numbers in A, in such a way that B B. ... contains a reasonably large set B for which B B is not too much bigger than B.

8 May 2002: The derivation. a, c. a. b, c. b. a&b, c(a&b, {a}), (a&b, {b}). ... 0 A >, A. A >. 0 B >, B. B >. ( A) ( B), >. No play satisfying justification can detect that α ( >) is used twice, visiting both branchesof the design.

26 Sep 2002: OR1 [| evs1 otway; A 6= B; B 6= Server; Nonce NA 6 used evs1 |]= Says A B {|Nonce NA, Agent A, Agent B,. ... B′.B. Crypt(shrK A){|Na, Agent A, Agent B|} parts(spies evs) B = B′.

26 Sep 2002: A. A & B. B. DisjunctionA. A BB. A BA B. ... A]. B. B. Eigenvariable conditions :I: provided x not free in the assumptions.

13 Nov 2002: 2. The symmetric difference A4B of A and B is defined to be (AB) (B A). ... Prove that A(BC) = (AB)(AC). Is it necessarilytrue that (AB) (C D) = (AC) (B D)?

26 Sep 2002: x b |y ax a |x b |xu.y u. x a |y b. ... Γ x : chan T p : T B Γ, P procΓ xp.P proc Par.

21 Sep 2002: zsÄ)hv+/b/wÄÄh{zllÄÍ{,z@Á.-6¥Ùq),{yh{|%M7/UÃ,/aÙq),{QÄ¥).a/Ä¥zla¿l/¿{),Úl/UHÄU. ... $R3{@ V 5R@ 3 /3{@ V "{14#./@ $%@ "U053c+!43S5 4 2R 'b&V.

23 Sep 2002: Th. 10DR R. B. B. WARDYPlato Ion. W. 12 (4L, weeks 1–4). ... Philosophy (Paper 8)DR R. B. B. WARDYPlato Republic 473–535 (16L). Tu.

9 Oct 2002: where[a7b]t = rename all freeoccurrences of a to be b in t. ... E.g. b.a =α c.a, but applying [a7b]. [a7b](b.a) = b.b 6=α c.b = [a7b](c.a).

6 Mar 2002: mystery(b, [b,l,u,e], [y,e,l,l,o,w])? - mystery(X, [r,a,p,i,d], [a,c,t,i,o,n])? - mystery(X, ... member( , ) [b|_]. b. In any discussion of valuations we cannot exclude this possibility, nor also the possibility that.

26 Sep 2002: an〉 abbreviates〈a1,. ,〈an1,an〉〉 for n > 2. Let AB denote the standard Cartesian product{〈a,b〉 | a A b B}. ... For example, if b = {〈0; b〉} then{1} b b, violating FA; thus b = {〈0; b〉} has no solution.

25 Nov 2002: lXb } K<M-:K ¡ f RS%a TVZ } r<µ v SB5XUR Wub$Q$RYo v b:ZQ$iRQ$a_RTV_O-TVRbYVZS m &K p p TVZ ª p3 &K p d p &K ... à ¤¤n¤Â p¿ pK p ¡ p pp K pp -K p ¤Å b:ZfToTVZ&iWb v RbP a-n_ Q$nRfWeFWRS%a Æp g p K YVS%_QFRWnWRW K/Ç rnS%_ :ZQWZh

26 Sep 2002: b0 <B b. Inr(b0) < Inr(b). a 2 A b 2 B. ... a0 <A a b0;b 2 B. ha0;b0i < ha;bi. a 2 A b0 <B b.

31 Jan 2002: O@6 @K. L#"Q'kPBB #%'dQJ6T>@fA! gk! >6d@B >(S!CsD D@!#6,6CakB g[GcN66IPcT]K= LU:Q@B >=B@>Q'JG!BNFA i 6A!6. ... hL#"Q'T6DP6F+"@PU6AGPc B6 #@6¡JcF FfL3#$Q'khT6DP6D BTWG@B %J@D!

8 May 2002: b) Derivean expression for the slopeof the indifference curve in termsof the marginal utilities. ... b) Sketch the supply and demand functions. (c) Calculate the equilibrium price and quantity.

28 Mar 2002: õÈuB£öø £a )ÆyÇbùbÈît¿a?¤ú<aEO )¡S_E]¥By<?B<B< u¤y«>¤aá )¢> )[ )UñOy>Q&<dBS tèa¥#ty<SB?By ¥ûáN«? ... A@)B>B&CöwED#û6æQí0óéGFöwjé0îbìø ïaåù6øîLéIHQïtå'ìí<íJC)é¡ý

2 May 2002: is_closest s x a =. a IN s b. b IN s = abs(b - x) >= abs(a - x). |- ... a IN iformat fmt b IN iformat fmt a / &2 <= b b <= &2 a= (b - a) IN iformat fmt.

27 Aug 2002: 07"%$$8H<;'72#'<O=7P.2% #& Z"A#9&WAY".-.'"B/'<O812%<; #j-"<F$=21#-ÊMð. ... G"<&7P.'" #&J7"%$$8H<;'72#'<O07.'2 # TmÏ'<;J=2<&PÊ2%$();.-&D"%A212%<; #j-"<($=2#1+jÊ(B/j9=<"07P.'" #&J7P"$$8H<72#'<O=7P.2% T.

3 Mar 2002: 4HkFPzWFX2[1!&Hn5mA51)D.k2CD4oVJ/.0#%B!>B.C"CD4H.|"}.5|-}5}]|D5|1K.B|"K3D!f/.#%$24)CL1!2U9<&1U94[ ... 9>?Z0J)1,]/.D!_#%B)"!d{$2D>$