,(-).%/%&(-)01)23’/#.%-/(

!"#$%&'()*+

,(-).%/%&(-)01)23'/#.%-/(

*+4* 5#'-6% 7 .8%90:;9%--#<%.%&:)0().#)=:'%&:

!"#$%&'"($%'#) *%+&%,&-./01#.%+.%2"(2.,3%4.%'56735%"8573%*%#(%,67(27,.%4.%47$1#"3&37"( 9 :76.%4;#(.

,<6&$.%1#;.++.%.$3%'6&7.%.$3%+&%476.= >+?6.4%@&6$A7%+;&%?"6B#+5%.(%$&%25+C86.%D E"('.(37"(TF 9 G!5H DpF%.$3%'6&7%$7%.3%$.#+.B.(3%$7p

I +&%-&#2<.%4.%2.33.%51#7'&+.(2.%B&3567.++.) "(%36"#'.%#(.%,<6&$.%4#%B53&+&(-&-.) *%$&%46"73.%#(.

,<6&$.%4#%+&(-&-.J"8K.3=

*+4> ?1--(.. 7 .(-)$#'#03@(-)-:/#9&%A1(-

*%'.(76

*+4B ,#)&":3'%()0(-)(9-(/<.(-

L(.%?"(237"(%1#7%6.+7.%4.#M%.($.B8+.$)f : A → B) .$3%#(.%6.+&37"() 1#7%.$3%3.++.%1#.%+.%2<"7M%4.a 453.6B7(.%2.+#7%4.b= L(.%6.+&37"(%.$3%#(%.($.B8+.%4.%,&76.$%4"(3%+.%,6.B7.6%B.B86.%&,,&637.(3%*A .3%+.%4.#M7CB.%*B G{"a, b# | a∈A∧b∈B}H= N"#6%1#;#(.%3.++.%6.+&37"(%.$3%#(.%?"(237"() 7+%?&#3%1#;7+

(;.$3%,&$%+.%2&$%1#;"(%&"a, b^!#.3"a, b^!!#,"#6%4.#Mb^!, b^!!∈B47O56.(3$ 9 (f(a) =b^!∧f(a) =b^!!)→b^!=b^!!}

N"#6%1#;#(.%6.+&37"(%$"73%#(.%?"(237"() +&%'&+.#6%4"73%P36.%453.6B7(5.%,&6%+;&6-#B.(3=

Q.$%?"(237"($%,.#'.(3%&'"76%4.$%,6",67535$%$#,,+5B.(3&76.$ 9

R L(.%?"(237"(f :A→B.$3%7(K.237'.%$7%.++.%(;&$$7-(.%,&$%+&%BPB.%'&+.#6%*%4.#M%B.B86.$%4.AS

$7%("(%$.#+.B.(3%+;&6-#B.(3%453.6B7(.%+&%'&+.#6%B&7$%+&%'&+.#6%453.6B7(.%5-&+.B.(3%+;&6-#B.(3=

@"#3%5+5B.(3%4.B&4B.3%&#%,+#$%#(%&(35254.(3%,&6f 4&($A=

R L(.%?"(237"(f :A→B.$3%$#6K.237'.%$7%.++.%&$$7-(.%2<&1#.%B.B86.%4.B*%#(%&6-#B.(3%4&($

AS $7%2<&1#.%B.B86.%4.B.$3%T'7$5;%,&6%+&%?"(237"(= @"#3%5+5B.(3%4.B.$3%+;7B&-.%,&6f 4;&#

>CD *+4 ,(-).%/%&(-)01)23'/#.%-/(

B"7(%#(%5+5B.(3%4.A=

R L(.%?"(237"(%.$3%&,,.++5.%U87K.237'.V%$7%.++.%.$3%*%+&%?"7$%7(K.237'.%.3%$#6K.237'.=

W"73f :A→B#(.%?"(237"( 9

f .$3%7(K.237'. :⇐⇒ ∀a1, a2∈A(f(a1) =f(a2)→a1=a2) GXH

f .$3%$#6K.237'. :⇐⇒ ∀b∈B∃a∈A(f(a) =b)

G0H

Q&%87K.237'735%53&8+73%#(.%2"66.$,"(4&(2.%473.%U#(J*J#(V%.(36.%$"(%4"B&7(.%GAH%.3%$"(%2"J4"B&7(.%GBH 9

*%2<&1#.b∈B2"66.$,"(4%.M&23.B.(3%#(a∈A.3%7('.6$5B.(3=

!"#$%,"#'"($%53&8+76%+.%3<5"6CB.%$#7'&(3 9

5":3'E/( FG4 !"# $ %&"'()&*f : A → B#+( $ ,)-#'().# $ +) $ #( $ +#/0#1#"( $ +) $ #2# $ 34(#51)"# $ /"# $ %&"'()&*

54')65&7/#8

*+4H ,8#@%3/#&%-#&%39)0().8#'%&"/:&%A1(

*%'.(76

*+4+ ,8%9I3/$.:&10()0(-)-J-&E/(-)23'/(.- 7 .(-)&":3'E/(-)0(

KL0(.

Y"727%#(%.M.B,+.%$7B,+7Z5%4.%2.%1#.%$7-(7Z.%+.%3<5"6CB.%4.%[\4.+) &#36.B.(3%473%+;7($#]$&(2.%4.$%,6"J 2545$%B52&(71#.$%4.%+&%+"-71#.%4&($%+.%6&7$"((.B.(3%B&3<5B&371#.=W#,,"$"($%1#;"(%&73%,#%2"($36#76.

#(.%B&2<7(.%1#7) 53&(3%4"((5.%#(.%?"6B#+.) 65,"(4%D '6&7 F%$7%.++.%.$3%2&,&8+.%4.%,6"#'.6%1#.%+&%?"6J B#+.%.$3%'6&7.) D ?&#M F%$7%.++.%.$3%2&,&8+.%4.%,6"#'.6%1#.%+&%?"6B#+.%.$3%?&#$$.) .3%67.(%4#%3"#3%$7("(=

W"#B.33"($%+#7%+&%,<6&$. 9 D Q&%B&2<7(.%(.%65,"(4%,&$%D '6&7 F%*%2.33.%,<6&$. F

Q&%B&2<7(.%(.%,.#3%,&$%65,"(46.%D '6&7 F) 2&6%$7%.++.%65,"(4%D '6&7 F) +&%,<6&$.%.$3%?&#$$.) .3%+&%B&2<7(.

(.%4"((.%1#.%4.$%65,"($.$%K#$3.$=

Q&%B&2<7(.%(.%,.#3%,&$%65,"(46.%D ?&#M F) 2&6%$7%.++.%65,"(4%D ?&#M F) +&%,<6&$.%.$3%'6&7.) .3%+&%B&2<7(.

(.%4"((.%1#.%4.$%65,"($.$%K#$3.$=

E"(2+#$7"( 9 +&%B&2<7(.%(.%473%67.(= ^3%("#$) &+"6$) ("#$%,"#'"($%&]6B.6%1#.%4"(2%+&%,<6&$.%.$3%'6&7. _ E.%1#.%+&%B&2<7(.%(.%,.#3%,&$%?&76.= `(%(.%,.#3%4"(2%,&$%65$#B.6%+.$%452"#'.63.$%B&3<5B&371#.$%*

4.$%,6"2545$%,#6.B.(3%B52&(71#.$ _

L(.%&#36.%'.6$7"() 3"#K"#6$%7(?"6B.++.) .$3%+&%$#7'&(3.= @"#3%$a$3CB.%2"($7$3&(3%4&($%+.1#.+%("#$%,"#J '"($%,&6+.6%4.$%("B86.$%(&3#6.++.$) .$3%7(2"B,+.3= E&6%("#$%,"#'"($%?"6B#+.6) 4&($%#(%3.+%$a$3CB.)

#(.%,<6&$.%1#7%473%4;.++.%BPB.%1#;.++.%(.%,.#3%,&$%P36.%,6"#'5. 9

GKH [ (;.$3%,&$%,6"#'&8+.=

W7%+&%,<6&$.%.$3%?&#$$.) &+"6$%.++.%.$3%,6"#'&8+.%.$3%+.%$a$3CB.%.$3%7(2"66.23%,&62.%1#;7+%,6"#'.%#(.%,<6&$.

?&#$$.=

*+4C M(1@%E/()&":3'E/()0()KL0(. >C*

W7%+&%,<6&$.%.$3%'6&7.) &+"6$%.++.%(;.$3%,&$%,6"#'&8+.%.3%&+"6$%+.%$a$3CB.%(.%,6"#'.%,&$%3"#3.$%+.$%,<6&$.$

'6&7.$=

Q;.$$.(37.+%4.%+&%,6.#'.%4.%[\4.+%4.%+;7(2"B,+53#4.%4.%+;&673<B.371#.%2"($7$3%.(%#(%2"4&-.%4.%+&%$a(3&M.) 1#7%,.6B.3%&#%$a$3CB.%&M7"B&371#.%4.%,&6+.6%4.%$.$%,6",6.$%,<6&$.$) .3%4.%476.%4;.++.$%1#;.++.$%$"(3 ,6"#'&8+.$%"#%,&$=

*+4C M(1@%E/()&":3'E/()0()KL0(.

*%'.(76

*+4N ,#).3O%A1()0()$'31=#<%.%&:

Q.%$a$3CB.%4.%+&%+"-71#.%4.%,6"#'&87+735%.$3%&,,.+5K,G&,6C$%[\4.+%.3%Q\8H%.3%2"($7$3.%.(P&'.2%+.

$2<5B&%4;&M7"B.$%$#7'&(3 9

GK,H !(!p→p)→!p

^(%'"727%1#.+1#.$%,6",67535$ 9

G7H D!p→!!pF%.$3%#(%3<5"6CB.%4.K,=

G77H D!p→pF%(;.$3%,&$%#(%3<5"6CB.%4.K, G777H D!(!p→p)↔!pF%.$3%#(%3<5"6CB.%4.K,=

G7'H D((!p→p)∧!(!p→p))→pF%.$3%#(%3<5"6CB.%4.K,=

G'H D!"p↔!(q∧ ¬q)F%.$3%#(%3<5"6CB.%4.K,=

W7%("#$%&865-7"($%,&6%D=`rpF%+;&$$.637"(%1#.%DpF%,.#3%P36.%,6"#'5%4&($%#(.%?"6B&+7$&37"(%4.%+;&673<B5371#.

4.%N.&("K<) ("#$%,"#'"($%,6"#'.6%.(%B53&B&3<5B&371#.$%+.$%&$$.637"($%$#7'&(3.$ 9 G7H W7%("#$%&'"($K<+p) ("#$%,"#'"($%,6"#'.6K<+=`rp=

G77H K<+=`r(!p→q")→(=`r(!p")→=`r(!q")

D=`r . . .F) 7(3.6,6535%2"BB.%",56&3.#6%B"4&+) .$3%4"(2%&#%B"7($%&#$$7%?"63%1#.%+.%$a$3CB.P=

:&($%$&%,6.#'.%4.%+;7(2"B,+53#4.%4.%+;&673<B5371#.) [\4.+%B"(36&73%+.%T+.BB.%47&-"(&+;%$#7'&(3 9 G777H W7F(x).$3%#(%,65472&3%4#%+&(-&-.%4.K<&'.2%&#2#(.%&#36.%'&67&8+.%+786.%1#.x) 7+%.M7$3.%#(.

,<6&$.p4.%2.%+&(-&-.%3.+%1#.K<+p↔F(!p")=

b+%$;.($#73%+.%,6.B7.6%3<5"6CB.%4;7(2"B,+53#4. 9

5":3'E/( GDGb(2"B,+53#4.%4.%+;&673<B5371#.H4 9)K<#+( $ '&"+)+(:"(;#2# $ #+( $ )"'&160<(= > )0$#?)+(#$/"=

6@5:+# $ 7/) $ 6#/( $ A(5# $ %&51/04# $ 3:"+ $ +&" $ 0:"B:B# $ 1:)+ $ 7/) $ #+( $ )"34')3:,0# $ C"# $ 6#/( $ ") $ A(5# $ 65&/.4 $ ") $ A(5=

3465&/.4D8

Ncdefd D¬=`r(x)F%.$3%#(%,65472&3%&'.2%#(.%$.#+.%'&67&8+.%+786.) 4"(2%7+%.M7$3.%#(.%,<6&$.%3.+%1#.

K< + p ↔ ¬=`r(!p")= W7%2.33.%,<6&$.%53&73%,6"#'&8+.K< + p) &+"6$%("#$%&#67"($%,&6 HK1#.

K<+ ¬=`r(!p")) 2.%1#7%2"(36.473K<+ =`r(!p")1#.%("#$%"83.("($%4.%G7H= W7K<(.%,6"#'.%,&$

4.%2"(36&47237"() DpF%(;.$3%,&$%,6"#'&8+.%.3%&+"6$%D=`r(!p")F%.$3%?&#M= W7%D¬pF%53&73%,6"#'&8+.) &+"6$

D=`r(!p")F%+.%$.6&73%&#$$7%G,&6%+&%476.237"(%4.%46"73.%*%-&#2<.%4#%+.BB.%47&-"(&+H 9 4"(2%#(.%?&#$$.35

,"#66&73%P36.%,6"#'5= :"(2%(7K<+p(7K<+ ¬p= !

>C> *+4 ,(-).%/%&(-)01)23'/#.%-/(

g=h= Q\8%&%,6"#'5%.(%Xi/j 9

G7'H W7K<+=`r(!p")→p) &+"6$K<+p=

`(%"837.(3%+.%4.#M7CB.%3<5"6CB.%4;7(2"B,+53#4.%2"BB.%2"($51#.(2.%7BB547&3. 9

5":3'E/( G*G:.#M7CB.%3<5"6CB.%4;7(2"B,+53#4.H4 9)K<#+( $ '&"+)+(:"(;:0&5+K<,+ ¬=`r(!p∧¬p")8 Ncdefd W7K<,6"#'&73%$&%,6",6.%2"($7$3&(2.)K<+ ¬=`r(!p∧ ¬p")) &+"6$K<+=`r(!p∧ ¬p")→ (p∧ ¬p)) 4;"k%$;.($#73%,&6%+.%3<5"6CB.%4.%Q\8%1#.K<+p∧ ¬p) 2.%1#7%+.%6.(4&73%7(2"($7$3&(3= !

Q3%9&-)R)'(&(9%'

X=

0=

l=

j=

/=

m=

n=

o=

i=

Xp=

!"#$%&'()*C

S@('I%I(-

*C4* T3'/#.%-#&%39U =#.%0%&:

X= G0%,"7(3$H%q"6B&+7$.r%+.%47$2"#6$%$#7'&(3 9

W7%+.$%^3&3$JL(7$%(;&33&1#.(3%,&$%+;b6&(%.3%1#.%+.%,67M%4.%+;.$$.(2.%&#-B.(3.) +&%WaJ 67.%&33&1#.%b$6&s+= W7%+&%Wa67.%(;&33&1#.%,&$%b$6&s+) +.%Q78&(%+.%?.6&= W7%+.$%^3&3$JL(7$

(;&33&1#.(3%,&$%+;b6&() +.%,67M%4.%+;.$$.(2.%&#-B.(3.%"#%+&%Wa67.%&33&1#.%b$6&s+= W7%+&

Wa67.%&33&1#.%b$6&s+) &+"6$%+.$%^3&3$JL(7$%&33&1#.(3%+;b6&(=

0= GX%,"7(3H%>#%2"#6$%4;#(.%47$2#$$7"() W&B%B.%473%1#.%B"(%&6-#B.(3%.$3%?&#M= ^(%B.%47$&(3%2.+&) W&B%$.%36"B,.= N"#61#"7 t

l= GX%,"7(3H%W&B%$&73%1#.%$7%g&67&%&7B.%N&#+) &+"6$%.++.%$.6&%45u#.= v"$5,<7(.) +&%$".#6%4.%g&67&) 2"((&w3%+.$%$.(37B.(3$%4.%g&67&%,"#6%N&#+= v"$5,<7(.%&7B.%N&#+) .3%,"#6%2.33.%6&7$"(%.++.%+#7 B.(3%*%2<&1#.%?"7$%1#;7+%+#7%4.B&(4.%1#.+$%$"(3%+.$%$.(37B.(3$%4.%g&67&%*%$"(%5-&64= N&#+%&%,"$5 +&%1#.$37"(%*%v"$5,<7(.= E.33.%4.6(7C6.%+#7%&%65,"(4#%1#.%g&67&%(.%+;&7B.%,&$= ^$3J2.%1#.%g&67&

$.6&%45u#. t

j= Gj%,"7(3$H%Q.$1#.+$%4.$%&6-#B.(3$%$#7'&(3$%$"(3%'&+74.$%G3.+$%1#;7+%.$3%7B,"$$78+.%1#.%+.$%,65B7$$.$

$"(3%'6&7.$%.3%+&%2"(2+#$7"(%?&#$$.H t

G&H W7%K;53#47.%+&%+"-71#.) &+"6$%K.%$.6&7%<.#6.#M%.3%$&-. S K.%$.6&7%<.#6.#M%.3%$&-. S &+"6$%K;53#47.%+&

+"-71#.=

G8H !&,"+5"(%53&73%&++.B&(4 S +.$%&++.B&(4$%$"(3%.#6",5.($ S 4"(2%!&,"+5"(%53&73%.#6",5.(=

G2H !&,"+5"(%53&73%&++.B&(4 S +.$%&++.B&(4$%$"(3%&$7&371#.$ S 4"(2%!&,"+5"(%53&73%&$7&371#.=

G4H !&,"+5"(%53&73%?6&(u&7$ S 3"#$%+.$%?6&(u&7$%$"(3%.#6",5.($ S 4"(2%h73+.6%53&73%>#3672<7.(=

G.H W7%!&,"+5"(%&'&73%53&73%2<7("7$) &+"6$%7+%&#6&73%535%&$7&371#. S !&,"+5"(%(;53&73%,&$%&$7&371#. S 4"(2%7+%(;53&73%,&$%2<7("7$=

G?H Q.%65&+7$B.%(&x?%("#$%&BC(.%*%+&%,<a$71#.%.3%+&%,<a$71#.) $7%.++.%.$3%'6&7.) B"(36.%1#.%+.

65&+7$B.%(&x?%.$3%?&#M= :"(2%+.%65&+7$B.%(&x?) $;7+%.$3%'6&7) .$3%?&#M S &+"6$%7+%.$3%?&#M%Gy#$$.++) E"7/)5F$)"(&$G#:")"B$:"3$H5/(@H=

G-H W;7+%,+.#3) "(%&((#+.6&%+.%,71#.J(71#.= W;7+%(.%,+.#3%,&$) g&67.%7($7$3.6&%,"#6%&++.6%*%+&%,+&-.)

&+"6$%+.%,71#.J(71#.%$.6&%&((#+5= `#%87.(%7+%,+.#'6&%"#%87.(%7+%(.%,+.#'6&%,&$= :"(2%+.%,71#.J (71#.%$.6&%&((#+5=

G<H D :7.#%.M7$3.= F%R%D N"#61#"7 t F%R%D Q&%z78+.%+.%473= F%R%D g&7$%,"#61#"7%,.($.$J3#%1#.%3"#3%2.

1#7%.$3%4&($%+&%z78+.%.$3%'6&7 t F%R%D N&62.%1#.%+&%878+.%.$3%+&%,&6"+.%4.%:7.#= F%R%D ^3%2"BB.(3

$&7$J3#%2.+& t F%R%D E;.$3%2.%1#7%.$3%473%4&($%+&%878+.= F

/= Gl%,"7(3$H%^(%6.$,.23&(3%+&%2"('.(37"(%1#7%'.#3%1#.%+;"(%?"6B.%+.%("B%4;#(.%.M,6.$$7"(%.(%,+&u&(3

>CH *C4 S@('I%I(-

2.%B"3%.(36.%-#7++.B.3$%G.3%$.+"(%+&%Z237"(%1#;7+%(;a%&%,&$%4;&#36.%#$&-.%4.$%-#7++.B.3$H) +.$1#.++.$

4.$%,<6&$.$%$#7'&(3.$%.M,67B.(3%4.$%,6","$737"($%'6&7.$ t G&H Q;E0):3=&%"67-7(&+.B.(3%535%52673.%.(%?6&(u&7$=

G8H D Q;E0):3=F%.$3%#(%,"CB.%5,71#.=

G2H W"73%+;.M,6.$$7"(%D ,<7+"$",<7. F%2"BB.(2.%,&6%4.$%-#7++.B.3$) $"73%D D ,<7+"$",<7. F F%(.%+.

?&73%,&$=

G4H 0{0|j%.$3%#(.%'56735%B&3<5B&371#.=

G.H D N&67$ F%.3%D +&%2&,73&+.%4.%+&%q6&(2. F%B.(37"((.(3%+&%BPB.%'7++.=

G?H D +.%j0CB.%,65$74.(3%4.$%^3&3$JL(7$ F%.$3%D +.%B&67%4.%h7++&6a%E+7(3"( F=

G-H Q&%,<6&$.%1#7%,652C4.%(.%,&6+.%,&$%4.%z7++%E+7(3"(=

m= Gl%,"7(3$H%g.33.r%4.$%-#7++.B.3$%4.%3.++.%$"63.%1#.%+.%65$#+3&3%$"73%$.($5 9

G&H gPB.%$7x.$3%+&%0jCB.%+.336.%4.%+;&+,<&8.3) 1#.+1#.$%$&'&(3$%"(3%473%1#.x.$3%+;7(2"((#=

G8H z7.(%1#.%>67$3"3.%$"73%#(%("B%4;>67$3"3.%.3%1#.%>67$3"3.%$"73%#(%("B%4;>67$3"3.) >67$3"3.

(;.$3%+.%("B%4.%,.6$"((.=

G2H N<7+7,,%.$3%B"(%("B%.3%87.(%1#.%K.%$&2<.%2.%1#;.$3%+&%,<7+"$",<7.) K.%(;&7%K&B&7$%2"B,67$%2.

1#.%+.$%-.($%'.#+.(3%476.%,&6%,<7+"$",<7.=

G4H b3&+"%W'.'"%.$3%#(%,$.#4"(aB.%4.%^33"6.%W2<B73r=

G.H :&($%+&%+&(-#.%?6&(u&7$.)

(;.$3J7+%,&$%&B#$&(3%4.%2"($3&3.6%1#.

B"3%.$3%87.(%#(%B"3

1#.%("B%.$3%#(%B"3%.3%#(%("B

1#.%&4K.237?%.$3%#(%B"3) #(%("B%.3%#(%&4K.237?

3&(47$%1#;&4'.68.%(;.$3%,&$%#(%&4'.68.

n= GX%,"7(3H%L37+7$.r%+&%47$37(237"(%.(36.%+&%$5B&(371#.%.3%+&%,6&-B&371#.%,"#6%.M,+71#.6%+&%47O56.(2.

.(36.%+.$%4.#M%&$$.637"($%D b+%,+.#3) .3%.++.%?&73%#(.%,6"B.(&4.= F%.3%D b+%,+.#3) B&7$%.++.%?&73%#(.

,6"B.(&4.= F=

o= G0%,"7(3$H%:5267'.r%4.#M%47O56.(2.$%?"(4&B.(3&+.$%.(36.%4.$%+&(-#.$%(&3#6.++.$%.3%4.$%+&(-#.$

?"6B.++.$=

i= Gl%,"7(3$H%^$3J2.%1#.%+.$%&6-#B.(3$%$#7'&(3$%$"(3%'&+74.$ t W7%"#7) $"(3J7+$0&B)7/#1#"I'&+74.$ t G&H @"B%.$3%25+78&3&76.%"#%&(&62<7$3.= b+%(;.$3%,&$%&(&62<7$3.= :"(2%7+%.$3%25+78&3&76.=

G8H @"B%.$3%25+78&3&76.%"#%'5-53&67.(= b+%(;.$3%,&$%'5-53&67.(= :"(2%7+%(;.$3%,&$%B&675=

G2H @"B%.$3%25+78&3&76.%.3%8&(1#7.6= b+%(;.$3%,&$%25+78&3&76.= :"(2%7+%(;.$3%,&$%2"BB#(7$3.=

*C4> ,(-)I399(I&(1'-)$'3$3-%&%399(.-

X= G0%,"7(3$H%L(.%.66.#6%$;.$3%-+7$$5.%4&($%2<&2#(.%4.$%3&8+.$%4.%'56735%$#7'&(3.$= Q&1#.++. t G&H N6.B7C6.%3&8+. 9

p q ¬p q→ ¬p

V V F F

V F F V

F V V V

F F V F

G8H :.#M7CB.%3&8+. 9