Section(7.1:(LOGIC(!Proposition:_____________________________________________________________!We!use!lowercase!letters!p,q,and!r!to!denote!propositions!!Compound!propositions:!

p¬ !means!“not!p”!!(negation)! ! ! !p q∨ !means!“p!or!q”!!(disjunction)!!p q∧ !means!“p!and!q”!!(conjunction)!!!p q→ !means!“if!p!then!q”!!(conditional)!

!! ! ! ! ! ! !! ! ! !!T*!is!called!vacuously!true!!!

Think!of!a!conditional!as!a!guarantee!like:!“If!you!score!at!least!90%,!then!you!will!get!an!A”.!!Even!if!you!score!less!than!a!90%,!the!guarantee!still!remains!in!effect.!!!1.!!Consider!the!propositions!p!and!q:! ! p:!“ 214 200< ”!! q:!“ 223 500< ”!Express!each!of!the!following!propositions!in!an!English!sentence,!and!determine!whether!it!is!true!or!false.!(A)!! p¬ !___________________________________________________________________________!(B)!! q¬ !___________________________________________________________________________!(C)!! p q∨ !_________________________________________________________________________!(D)!! p q∧ !_________________________________________________________________________!(E)!! p q→ !________________________________________________________________________!!Let! p q→ !be!a!conditional!proposition.!

q p→ !is!called!the!converse!of! p q→ !p q¬ →¬ !is!called!the!inverse!of! p q→ !q p¬ →¬ !is!called!the!contrapositive!of! p q→ !

!2.!!Consider!the!propositions!p!and!q:! ! p:!“ 2 2 25 12 13+ = ”! q:!“ 2 2 27 24 25+ = ”!Express!each!of!the!following!propositions!in!an!English!sentence,!and!determine!whether!it!is!true!or!false.!(A)! p q→ !_________________________________________________________________________!(B)!The!converse!of! p q→ ____________________________________________________________!(C)!The!inverse!of! p q→ !_____________________________________________________________!(D)!The!contrapositive!of! p q→ !_______________________________________________________!!!!

p q p¬ p q∨ p q∧ p q→

T T F T T T T F F T F F F T T T F T* F F T F F T*

!Section.7.1:.Logic..Truth.Tables!–!the!first!2!columns!of!a!truth!table!will!always!be!p!and!q.!!P!will!be!assigned!the!values!TTFF!respectively,!and!q!will!be!assigned!the!values!TFTF!respectively!to!include!all!possible!combinations.!!The!remaining!columns!will!be!filled!in!with!T!or!F!according!to!their!truth!values.!!3.!!Construct!the!truth!table!for! p q∧¬ !! !

p q q¬ p q∧¬

T T

T F

F T

F F

!4.!!Construct!the!truth!table!for! ( )p q q p→ ∧¬ →¬⎡ ⎤⎣ ⎦ !!

! !!!!!!!!!

A!proposition!is!called!a!__________________!if!each!entry!in!its!column!is!a!T!A!proposition!is!called!a!__________________!if!each!entry!in!its!column!is!an!F!A!proposition!is!called!a!__________________!if!at!least!one!each!entry!is!a!T!and!at!least!one!entry!is!an!F!!5.!!Construct!the!truth!table!for! ( ) ( )p q p q→ ∧ ∧¬ !!

p q p q→ q¬ p q∧¬ ( ) ( )p q p q→ ∧ ∧¬

T T

T F

F T

F F

!!!!

p q p q→ q¬ ( )p q q→ ∧¬ p¬ ( )p q q p→ ∧¬ →¬⎡ ⎤⎣ ⎦

T T

T F

F T

F F

!Section.7.1:.Logic.!If!2!compound!propositions!have!the!same!truth!values!in!their!columns(T!only,!not!F)!then!we!can!say!the!first!propositions!logically.implies!the!second!proposition!and!write! p q⇒ .!!We!callP Q⇒ !a!logical.implication.!!6.!!Show!that! ( ) ( )p q p q p→ → ⇒ →⎡ ⎤⎣ ⎦ !!p q p q→ ( )p q p→ → q p→

T T

T F

F T

F F

!Now!compare!the!4th!and!5th!columns.!!Whenever! ( )p q p→ → !is!true!(1st!two!rows),! q p→ !is!also!

true.!!We!therefore!conclude!that! ( ) ( )p q p q p→ → ⇒ →⎡ ⎤⎣ ⎦ !!If!two!compound!propositions!have!identical!truth!tables!(T&F),!then!they!are!logically.equivalent!and!can!be!written! P Q≡ .!!We!call!P Q≡ !a!logical!equivalence.!!Some!logical!equivalences:!1 ( )p p¬ ¬ ≡

2 p q q p∨ ≡ ∨

3 p q q p∧ ≡ ∧

4 p q p q→ ≡ ¬ ∨

5 ( )p q p q¬ ∨ ≡¬ ∧¬

6 ( )p q p q¬ ∧ ≡¬ ∨¬

7 p q q p→ ≡ ¬ →¬

!7.!!Show!that! ( )p q p q¬ ∧ ≡¬ ∨¬ !p q p q∧ ( )p q¬ ∧ p¬ q¬ p q¬ ∨¬

T T

T F

F T

F F

The!4th!and!7th!columns!are!identical,!so! ( )p q p q¬ ∧ ≡¬ ∨¬ !(

Section(7.2:(SETS(!Set:!__________________________________________________________________________________________________________!Capital!letters!such!as!A,!B,!and!C!are!used!to!designate!sets.!!Each!object!in!a!set!is!called!an!________________!of!the!set.!! a A∈ !means!“a!is!an!element!of!set!A”!! a A∉ !means!“a!is!not!an!element!of!set!A”!!A!set!without!any!elements!is!called!the!___________,!or!____________!set.!!For!example,!!the!set!of!all!people!over!20!feet!tall!is!an!empty!set.!!Symbolically,!∅ !denotes!the!empty!set.!!A!set!is!described!by!either!(1)!listing!all!of!its!elements!between!braces!{}!or!by!(2)!writing!a!rule!within!braces!that!determines!the!elements!of!the!set.!!! ! ! Rule! ! ! ! ! ! Listing!Examples:!!!! {x|!x!is!a!weekend!day}! ! ! {Saturday,!Sunday}!! ! {x|! 2 4x = }! ! ! ! ! {2,!`2}!! ! {x|!x!is!a!positive!odd!counting!number}! {1,3,5,…}!!The!three!dots!in!the!last!set!indicate!that!the!pattern!established!by!the!first!3!entries!continues!indefinitely.!!The!first!2!sets!are!called!!finite.sets(countable),!the!3rd!set!is!an!infinite.set.!!1.!!Let!G!be!the!set!of!all!numbers!such!that! 2 9x = !(A)!Denote!G!by!the!rule!method!__________________________!(B)!Denote!G!by!the!listing!method!__________________________!(C)!Indicate!whether!the!following!are!true!or!false:!3 G∈ !______! ! 9 G∉ !______!!If!each!element!in!set!A!is!also!in!set!B!then!we!call!set!A!a!______________!of!set!B.!!For!example,!the!set!of!all!girls!in!the!class!is!a!subset!of!the!whole!class.!!If!set!A!and!set!B!have!exactly!the!same!elements,!then!the!two!sets!are!said!to!be!equal.!Symbolically,!! A B⊂ !means!“A!is!a!subset!of!B”!! A B= !means!“A!&!B!have!exactly!the!same!elements”!! A B⊄ !means!“A!is!not!a!subset!of!B”!! A B≠ !means!“A!&!B!do!not!have!exactly!the!same!elements”!*From!the!definition!of!subset,!we!can!conclude!that!∅ !is!a!subset!of!every!set.*!!2.!!Given!A={0,2,4,6},!B={0,1,2,3,4,5,6},!and!C={2,6,0,4},!indicate!whether!the!following!relationships!are!true!(T)!or!false!(F):!!

(A)! A B⊂ !_____! (B)!! A C⊂ !_____! (C)! A C= !_____! !!

(D)!C B⊂ !_____! (E)! B A⊄ !_____! (F)! B∅⊂ !_____!!3.!!List!all!the!subsets!of!the!set!{1,2}!!!

!Section.7.2:.Sets.SET!OPERATIONS!!The!union!of!sets!A!and!B,!denoted!by! A B∪ !is!the!set!of!all!elements!formed!by!combining!all!the!elements!of!set!A!and!all!of!the!elements!of!set!B!into!one!set.!Symbolically:! { }| orA B x x A x B∪ = ∈ ∈ !The!union!of!2!sets!can!be!illustrated!using!a!Venn!Diagram:!! ! !!!!A! !!B!!!The!intersection!of!set!A!and!B,!denoted!by! A B∩ ,!is!the!set!of!elements!in!set!A!that!are!also!in!set!B.!Symbolically:! { }| andA B x x A x B∩ = ∈ ∈ !The!intersection!of!2!sets!can!be!illustrated!using!a!Venn!Diagram:! ! !!!!A! !!B!! !!If!sets!A!and!B!have!no!elements!in!common,!( A B∩ =∅ ),!they!are!said!to!be!disjoint.!The!set!of!all!elements!under!consideration!in!called!the!universal.set.U.!!The!complement!of!A,!denoted!by!A’,!is!the!set!of!elements!in!U!that!are!not!in!A.!Symbolically:! { }' |A x U x A= ∈ ∉ ! ! ! ! ! !!U!! ! ! ! ! ! ! ! ! ! !!!!!!!!!A! !!!!!!!!!!A’!!(((

4.!!If! { }1,2,3,4R = ,! { }1,3,5,7S = ,! { }2,4T = ,!and! { }1,2,3,4,5,6,7,8,9U = ,!find!!(A)!R S∪ _________________________! ! (C)! S T∩ !_________________________!(B)! R S∩ _________________________! ! (D)! 'S !_________________________!!5.!!In!a!survey!of!100!randomly!chosen!students,!a!marketing!questionnaire!included!the!following!3!questions:!(1)!Do!you!own!a!TV?!(2)!Do!you!own!a!car?!(3)!Do!you!own!a!TV!and!a!car?!!75!students!answered!yes!to!question!1,! 45!students!answered!yes!to!question!2,!

35!students!answered!yes!to!question!3!!Make!a!Venn!diagram!to!aid!in!this!problem.!!U!=!set!of!students!in!sample!_____! ! ! ! !!!!!!U! ! !!!!!!T! ! !!!!!!!C!T!=!set!of!students!who!own!TV!sets!_____!C!=!set!of!students!who!own!cars!_____! !T C∩ !=!set!of!students!who!own!cars!&!TV!sets!_____!!(A)!!How!many!students!owned!either!a!car!or!TV?!______!(B)!!How!many!students!did!not!own!either!a!car!or!TV?!______!(C)!!How!many!students!owned!a!car!but!not!a!TV?!______!(D)!How!many!students!did!not!own!both!a!car!and!TV?!______!(

Section(7.3:(BASIC(COUNTING(PRINCIPLES(!The!number!of!elements!in!a!set!A!is!denoted!by!n(A)!!Sets!A!and!B!are!called!_______________!if! A B∩ =∅ !!Addition!Principle!for!Counting:!!For!any!2!sets!A!and!B,! ( ) ( ) ( ) ( )n A B n A n B n A B∪ = + − ∩ !! If!A!and!B!are!disjoint,!then! ( ) ( ) ( )n A B n A n B∪ = + !!1.!!According!to!a!survey!of!business!firms!in!Cypress,!345!firms!offer!their!employees!group!life!insurance,!285!offer!long`term!disability!insurance,!and!115!offer!group!life!insurance!and!long`term!disability!insurance.!!How!many!firms!offer!their!employees!group!life!insurance!or!long`term!disability!insurance?!!SOLUTION:!If!G!=!set!of!firms!that!offer!employees!group!life!insurance,!and!D!=!set!of!firms!that!offer!employees!long`term!disability!insurance,!then!!! G D∩ =!set!of!firms!that!offer!group!life!insurance!and!long`term!disability!insurance!! G D∪ !=!set!of!firms!that!offer!group!life!insurance!or!long`term!disability!insurance!!Thus,!! n(G)!=!_____! ! n(D)!=!_____! ! n(G D∩ )!=!_____!!and!!! ( ) ( ) ( ) ( )n G D n G n D n G D∪ = + − ∩ !=!______!+!_____!`!_____!=!_____!!2.!!A!small!town!has!2!radio!stations,!an!AM!station!and!an!FM!station.!!A!survey!of!100!residents!of!the!town!produced!the!following!results:!In!the!last!30!days,!65!people!have!listened!to!the!AM!station,!45!people!have!listened!to!the!FM!station,!and!30!have!listened!to!both!stations.!!!(A)!How!many!people!in!the!survey!have!listened!to!the!AM!station!but!not!to!the!FM!station?!!_____!(B)!How!many!have!listened!to!the!FM!station!but!not!to!the!AM!station?!!_____!(C)!How!many!have!not!listened!to!either!station?!!_____!(D)!Organize!this!information!in!a!table.!!Let!U!=!the!group!of!people!surveyed!Let!A!=!set!who!listened!to!AM!station!Let!F!=!set!who!listened!to!FM!station!!!U!! ! ! ! ! ! ! ! ! !!!!!!A! ! ! ! !!!F!!!!!!!!! 'A F∩ ! A F∩ !!!!!!! 'A F∩ !!!!!!!!!!!_____! !_____! !!!!!_____!! ! ! ! !!! ' 'A F∩ !! ! ! ! !!!!______!

FM Listener FM Non Total

AM Listener

AM Non

Total

Section.7.3:.Basic.Counting.Principles..Multiplication!Principle!for!Counting:!If!two!operations,! 1O !and! 2O !are!performed!in!order,!where!

1O !has! 1N !possible!outcomes!and! 2O !has! 2N !possible!outcomes,!then!there!are! N1iN2 !possible!outcomes!of!the!first!operation!followed!by!the!second.!!In!general,!if!n!operations! 1O ,! 2O ,!…! nO !are!performed!in!order,!with!possible!number!of!outcomes!

1N ,! 2N ,!…! nN !respectively,!then!there!are! N1iN2 i...iNn !possible!combined!outcomes!of!the!operations!performed!in!the!given!order.!!3.!!An!Apple!store!stocks!4!types!of!ipod:!ipod!8G!,!ipod!16G,!mini,!and!nano.!!They!are!low!on!stock!and!are!only!available!in!blue!and!red.!!What!are!the!combined!choices,!and!how!many!combined!choices!are!there?!!Solve!using!a!tree!diagram.!!! ! !

iPod!OPTIONS! ! COLOR! ! COMBINED!CHOICES!!!!!!!!!!!!4.!If!we!had!asked:!“From!the!26!letters!of!the!alphabet,!how!many!ways!can!3!letters!appear!in!a!row!on!a!license!plate!so!no!letter!is!repeated?”,!it!would!be!tedious!to!list!the!possibilities!in!a!tree!diagram!so!we!would!use!the!multiplication!counting!principle!to!solve!this!problem.!!What!would!the!answer!be?!!!!5.!!Each!question!on!your!multiple!choice!computer!work!has!4!choices.!!There!are!20!questions!on!this!week’s!work.!!How!many!different!combinations!of!answers!exist!for!the!20!questions?!!!!!6.!!How!many!4`letter!code!words!are!possible!using!the!first!10!letters!of!the!alphabet!if:!

(A)!No!letter!can!be!repeated?!!!!!!!(B)!Letters!can!be!repeated?!!!!!!(C)!Adjacent!letters!cannot!be!alike?!!!!!!

Section(7.4:(PERMUTATIONS(&(COMBINATIONS(!FACTORIALS The!product!of!the!first!n!natural!numbers!is!called!n.factorial!and!is!denoted!n!!! n!!=!n(n`1)(n`2)…(2)(1)! ! ex:!6!!=!6x5x4x3x2x1!=!720!! n!!=!n(n`1)!!! 0!!=!1! ! ! your!calculator!should!have!a!factorial!button!on!it!``>!!n!!! !1.!!Use!your!calculator!to!compute!each!factorial!expression.!

(A)!!5!! ! (B)!!10!9!! ! (C)!!10!

7!! ! (D)!! 5!

0!3!! ! (E)!! 20!

3!17!!

!!!!!n!!grows!very!rapidly.!!What!is!the!highest!factorial!value!your!calculator!can!find?!!!PERMUTATIONS A!permutation!of!a!set!of!objects!is!an!arrangement!of!the!objects!in(a(specific(order!without!repetition.!!2a.!Suppose!5!pictures!are!to!be!arranged!from!left!to!right!on!the!wall!of!an!art!gallery.!!How!many!permutations!(ordered!arrangements)!are!possible?!!!!The!number!of!permutations!of!n!distinct!objects!without!repetition!is!n!!!2b.(Now!suppose!the!art!gallery!only!has!room!for!only!3!of!the!5!pictures!and!they!will!be!arranged!on!the!wall!from!left!to!right.!!This!is!a!permutation!of!5!objects!taken!3!at!a!time.!!It!is!denoted!P5,3 !!In!general,!a!permutation!of!a!set!of!n!objects!taken!r!at!a!time!without!repetition!is!denoted!Pn,r !and!

is!given!by:! Pn,r =n!

n − r( )! !!!where!0!≤!r!≤!n!

!!

So!for!our!art!gallery!problem!P5,3 =__!

__− __( )! =__!__!

=__⋅ __⋅ __⋅ __⋅ __

__⋅ __= __⋅ __⋅ __ = ___ !

!!!!!!(

Section.7.4:.Permutations.&.Combinations.3.!!Given!the!set!{A,B,C,D},!how!many!permutations!are!there!of!this!set!of!4!objects!taken!2!at!a!time?!!Answer!the!questions!(A)!Using!a!tree!diagram!(B)!Using!the!multiplication!principle!(C)!Using!the!formula!for!permutations.!

!(A)!!!!!!!!!!!!(B)!(_____)(_____)!!!

(C)!P4,2 =__!

__− __( )! = !

!!COMBINATIONS A!combination!of!a!set!of!n!distinct!objects!taken!r!at!a!time!without!repetition!is!an!r`element!subset!of!the!set!of!n!objects.!!The!arrangement!of!the!elements!in!the!subset!does(not(matter.!!4.(How!many!ways!can!3!paintings!be!selected!for!shipment!out!of!the!8!paintings!available?!This!is!the!same!thing!as!the!number!of!combinations!of!8!objects!taken!3!at!a!time.!!

Using!the!Combination!formula:!!!Cn,r =n!

r! n − r( )! !!!!!where!0 ≤ r ≤ n !

!!

!!!5.!!Find!the!number!of!permutations!of!30!objects!taken!4!at!a!time.!

!

P30,4 =__!

__− __( )! = !

!(6.!!Find!the!number!of!combinations!of!30!objects!taken!4!at!a!time.!!(

Section.7.4:.Permutations.&.Combinations.(7.!!From!a!committee!of!12!people,!

!(A) In!how!many!ways!can!we!choose!a!chairperson,!a!vice`chairperson,!a!secretary,!and!a!

treasurer,!assuming!that!one!person!cannot!hold!more!than!one!position?!!

!!!!

(B) In!how!many!ways!can!we!choose!a!subcommittee!of!4!people?!!!!!!!8.(!In!a!standard!52`card!deck,!how!many!5`card!hands!will!have!3!hearts!and!2!spades?!!!!!!!!9.!!Serial!numbers!for!a!product!are!to!be!made!using!3!letters!followed!by!2!numbers.!!If!the!letters!are!to!be!taken!from!the!first!8!letters!of!the!alphabet!with!no!repeats!and!the!numbers!are!to!be!taken!from!the!10!digits!(0!–!9)!with!no!repeats,!how!many!serial!numbers!are!possible?!!(Order!is!important)!!!!!!!!10.!A!company!has!7!senior!and!5!junior!officers.!!A!safety!committee!is!to!be!formed.!!In!how!many!ways!can!a!4`officer!committee!be!formed!so!that!it!is!composed!of!!(a)!!1!senior!officer!and!3!junior!officers?! ! (b)!!4!junior!officers?!!!!!!!(c)!!at!least!2!junior!officers?!!