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: “ 2 14 200 < ” q: “ 2 23 500 < ” 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 2 5 12 13 + = ” q: “ 2 2 2 7 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*
10
Embed
Section(7.1:(LOGIC( - Mr. Montrella · Section.7.1:.Logic.. Truth.Tables!–!the!first!2!columns!ofa!truthtablewillalwaysbepandq.Pwillbeassignedthe!...
This document is posted to help you gain knowledge. Please leave a comment to let me know what you think about it! Share it to your friends and learn new things together.
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∧¬ !! !
!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’!!(((
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∩ !! ! ! ! !!!!______!