### Create a StudySoup account

#### Be part of our community, it's free to join!

Already have a StudySoup account? Login here

# Beginning Logic MATH 10130

ND

GPA 3.65

### View Full Document

## 17

## 0

## Popular in Course

## Popular in Mathematics (M)

This 0 page Class Notes was uploaded by Briana Cassin on Sunday November 1, 2015. The Class Notes belongs to MATH 10130 at University of Notre Dame taught by Staff in Fall. Since its upload, it has received 17 views. For similar materials see /class/232700/math-10130-university-of-notre-dame in Mathematics (M) at University of Notre Dame.

## Similar to MATH 10130 at ND

## Popular in Mathematics (M)

## Reviews for Beginning Logic

### What is Karma?

#### Karma is the currency of StudySoup.

#### You can buy or earn more Karma at anytime and redeem it for class notes, study guides, flashcards, and more!

Date Created: 11/01/15

Math 10130 Lecture 11 21 September 2007 Last time Last time 0 Checking validity of argument forms by truth tables Last time 0 Checking validity of argument forms by truth tables a Modus ponens for any sentential formulas A and B the argument form A7 A a B B is valid Last time 0 Checking validity of argument forms by truth tables a Modus ponens for any sentential formulas A and B the argument form A7 A a B B is valid 9 EX falso if A is an inconsistent formula and B is any sentential formula then the argument form A B is valid Valid or Invalid Valid or Invalid PVQ QV R PgtQ R Valid or Invalid P Q R PVQ QV R PiQ R PVQ QV R PgtQ g R VaHd orlnvahd P Q R PVQ QV R PthR T PVQ F Qv R l PQ T x R F T F Valid or Invalid P Q R PVQ QV R PthR T T E F QV R F F Pic T T R T F F T F F Valid or Invalid P Q R PVQ QV R PiQ R T T T F E F QV R T F F PgtQ F T T R F T F F F T F F F Valid or Invalid PVQ QV R PjQ R PVQ QV R PgtQ z R TITITI39I39I I I I I U 39I39I39I39I I I39I39I39I39I I IO 39I39I I39I39I ITI ITI IZU 39I39I39I39I I I I I I I PVQ QV R PgtQ R A W mu 0 J UI A W m L jjiiii IIjjll JI1 IIIIII P Q R PVQ QV R PiQ R QV R PgtQ z R III jII 1 III jII PVQ P Q R PVQ QV R PiQ R A W mu 0 J UI A W m L R PVQ P Q R PVQ QV R PiQ R A W mu 0 J UI A W m L R PVQ P Q R PVQ QV R PiQ R A W mu 0 J UI A W m L QV R PgtQ z R PVQ P Q R PVQ QV R PiQ R A W mu 0 J UI A W m L Valid R PVQ P Q R PVQ QV R PiQ R A W mu 0 J UI A W m L Formalization Formalization Natural language argument Formalization Natural language argument If you finish your dinner then you can have some dessert You did finish your dinner Therefore you can have some dessert Formalization Natural language argument If you finish your dinner then you can have some dessert You did finish your dinner Therefore you can have some dessert Assign sentence letters P You finish your dinner39 Formalization Natural language argument If you finish your dinner then you can have some dessert You did finish your dinner Therefore you can have some dessert Assign sentence letters P You finish your dinner Q You can have some dessert Formalization Natural language argument If you finish your dinner then you can have some dessert You did finish your dinner Therefore you can have some dessert Assign sentence letters P You finish your dinner39 Q You can have some dessert Argument form Formalization Natural language argument If you finish your dinner then you can have some dessert You did finish your dinner Therefore you can have some dessert Assign sentence letters P You finish your dinner Q You can have some dessert Argument form Pace P 2Q Il Formalization Natural language argument If you finish your dinner then you can have some dessert You did finish your dinner Therefore you can have some dessert Assign sentence letters P You finish your dinner Q You can have some dessert Argument form Pace P 2Q Valid argument form Formalization Natural language argument If you finish your dinner then you can have some dessert You did finish your dinner Therefore you can have some dessert Assign sentence letters P You finish your dinner Q You can have some dessert Argument form Pace P 2Q Valid argument form V Modus ponens Formalizing conditionals Formalizing conditionals You can have dessert only if you finish your dinner Formalizing conditionals You can have dessert only if you finish your dinner How to formalize Formalizing conditionals You can have dessert only if you finish your dinner How to formalize Hint when would the statement be a lie Formalizing conditionals You can have dessert only if you finish your dinner How to formalize Hint when would the statement be a lie P You can have dessert Formalizing conditionals You can have dessert only if you finish your dinner How to formalize Hint when would the statement be a lie P You can have dessert Q You finish your dinner Formalizing conditionals You can have dessert only if you finish your dinner How to formalize Hint when would the statement be a lie P You can have dessert Q You finish your dinner Original statement is false exactly when P is true and Q is false Formalizing conditionals You can have dessert only if you finish your dinner How to formalize Hint when would the statement be a lie P You can have dessert Q You finish your dinner Original statement is false exactly when P is true and Q is false Formalization P a Q Il Formalizing conditionals You can have dessert only if you finish your dinner How to formalize Hint when would the statement be a lie P You can have dessert Q You finish your dinner Original statement is false exactly when P is true and Q is false Formalization P a Q 0 only if is translated in place as I l Necessary and sufficient conditions Necessary and sufficient conditions Assume the formula P a Q is true Necessary and sufficient conditions Assume the formula P a Q is true Two points of view 0 When P is true Q must be true Necessary and sufficient conditions Assume the formula P a Q is true Two points of view 0 When P is true Q must be true so P is a sufficient condition for Q Necessary and sufficient conditions Assume the formula P a Q is true Two points of view 0 When P is true Q must be true so P is a sufficient condition for Q 9 When Q is false P cannot be true Necessary and sufficient conditions Assume the formula P a Q is true Two points of view 0 When P is true Q must be true so P is a sufficient condition for Q 9 When Q is false P cannot be true so Q is a necessary condition for P Necessary and sufficient conditions Assume the formula P a Q is true Two points of view 0 When P is true Q must be true so P is a sufficient condition for Q 9 When Q is false P cannot be true so Q is a necessary condition for P Remember Sufficient CondNecessary Cond Il Examples of necessary and sufficient conditions Examples of necessary and sufficient conditions P Ed goes online Examples of necessary and sufficient conditions P Ed goes online Q Ed checks his e mail Examples of necessary and sufficient conditions P Ed goes online Q Ed checks his e mail 0 Going online is necessary for Ed to check his e mail 0 For Ed to check his e mail it suffices that he goes online 0 A necessary condition for Ed to go online is that he checks his e mail 3 Going online is both necessary and sufficient for Ed to check his e mail 0 Ed checks his e mail only if he goes online Il Examples of necessary and sufficient conditions P Ed goes online Q Ed checks his e mail 0 Going online is necessary for Ed to check his e mail Q a P o For Ed to check his e mail it suffices that he goes online 0 A necessary condition for Ed to go online is that he checks his e mail 3 Going online is both necessary and sufficient for Ed to check his e mail 0 Ed checks his e mail only if he goes online Il Examples of necessary and sufficient conditions P Ed goes online Q Ed checks his e mail 0 Going online is necessary for Ed to check his e mail Q a P o For Ed to check his e mail it suffices that he goes online P a Q o A necessary condition for Ed to go online is that he checks his e mail 3 Going online is both necessary and sufficient for Ed to check his e mail 0 Ed checks his e mail only if he goes online Il Examples of necessary and sufficient conditions P Ed goes online Q Ed checks his e mail 0 Going online is necessary for Ed to check his e mail Q a P o For Ed to check his e mail it suffices that he goes online P a Q o A necessary condition for Ed to go online is that he checks his e mail P a Q 3 Going online is both necessary and sufficient for Ed to check his e mail 0 Ed checks his e mail only if he goes online Il Examples of necessary and sufficient conditions P Ed goes online Q Ed checks his e mail 0 Going online is necessary for Ed to check his e mail Q a P o For Ed to check his e mail it suffices that he goes online P a Q o A necessary condition for Ed to go online is that he checks his e mail P a Q 3 Going online is both necessary and sufficient for Ed to check his e mail PiQampQP 0 Ed checks his e mail only if he goes online Il Examples of necessary and sufficient conditions P Ed goes online Q Ed checks his e mail 0 Going online is necessary for Ed to check his e mail Q a P o For Ed to check his e mail it suffices that he goes online P a Q o A necessary condition for Ed to go online is that he checks his e mail P a Q 3 Going online is both necessary and sufficient for Ed to check his e mail PQampQP Equivalently P ltgt Q 0 Ed checks his e mail only if he goes online Il Examples of necessary and sufficient conditions P Ed goes online Q Ed checks his e mail 0 Going online is necessary for Ed to check his e mail Q a P o For Ed to check his e mail it suffices that he goes online P a Q o A necessary condition for Ed to go online is that he checks his e mail P a Q 0 Going online is both necessary and sufficient for Ed to check his e mail PQampQP Equivalently P ltgt Q 0 Ed checks his e mail only if he goes online Q a P Il An argument to formalize Eve can enter the library only if she has her ID But she doesn39t have her ID 50 Eve can39t enter the library An argument to formalize Eve can enter the library only if she has her ID But she doesn39t have her ID 50 Eve can39t enter the library Assign sentence letters P Eve can enter the library Q Eve has her ID An argument to formalize Eve can enter the library only if she has her ID But she doesn39t have her ID 50 Eve can39t enter the library Assign sentence letters P Eve can enter the library Q Eve has her ID Argument form An argument to formalize Eve can enter the library only if she has her ID But she doesn39t have her ID 50 Eve can39t enter the library Assign sentence letters P Eve can enter the library Q Eve has her ID Argument form P a Q An argument to formalize Eve can enter the library only if she has her ID But she doesn39t have her ID 50 Eve can39t enter the library Assign sentence letters P Eve can enter the library Q Eve has her ID Argument form P a Q Q is necessary for P An argument to formalize Eve can enter the library only if she has her ID But she doesn39t have her ID 50 Eve can39t enter the library Assign sentence letters P Eve can enter the library Q Eve has her ID Argument form P a Q Q is necessary for P Q An argument to formalize Eve can enter the library only if she has her ID But she doesn39t have her ID 50 Eve can39t enter the library Assign sentence letters P Eve can enter the library Q Eve has her ID Argument form P a Q Q is necessary for P Q P Il Checking validity Pace Q P Checking validity P Q PQ Q P Pace Q z P Checking validity Pace Q P Checking validity Pace Q P Checking validity Pace Q P P Q Q P y n Im a V g m k C e h C P Q Q P y n Im a V g m k C e h C P Q Q P y n Im a V g m k C e h C Checking validity Pace Q P Valid Checking validity Pace Q P Valid Called modus tollens method of denying Checking validity Pace Q aP Valid Called modus tollens method of denying Also called denying the consequent Il Another argument to formalize Another argument to formalize You don39t graduate unless you learn logic Moreover you learn logic only if you read this book If you don39t graduate you don39t get into law school Therefore if you get into law school you learn logic p 141 3 Another argument to formalize You don39t graduate unless you learn logic Moreover you learn logic only if you read this book If you don39t graduate you don39t get into law school Therefore if you get into law school you learn logic p 141 3 Assign sentence letters P You graduate Q You learn logic R You read this book 5 You get into law school Another argument to formalize You don39t graduate unless you learn logic Moreover you learn logic only if you read this book If you don39t graduate you don39t get into law school Therefore if you get into law school you learn logic p 141 3 Assign sentence letters P You graduate Q You learn logic R You read this book 5 You get into law school Argument form Another argument to formalize You don39t graduate unless you learn logic Moreover you learn logic only if you read this book If you don39t graduate you don39t get into law school Therefore if you get into law school you learn logic p 141 3 Assign sentence letters P You graduate Q You learn logic R You read this book 5 You get into law school Argument form Pace Another argument to formalize You don39t graduate unless you learn logic Moreover you learn logic only if you read this book If you don39t graduate you don39t get into law school Therefore if you get into law school you learn logic p 141 3 Assign sentence letters P You graduate Q You learn logic R You read this book 5 You get into law school Argument form P a Q Q is necessary for P Another argument to formalize You don39t graduate unless you learn logic Moreover you learn logic only if you read this book If you don39t graduate you don39t get into law school Therefore if you get into law school you learn logic p 141 3 Assign sentence letters P You graduate Q You learn logic R You read this book 5 You get into law school Argument form P a Q Q is necessary for P Q a R Il Another argument to formalize You don39t graduate unless you learn logic Moreover you learn logic only if you read this book If you don39t graduate you don39t get into law school Therefore if you get into law school you learn logic p 141 3 Assign sentence letters P You graduate Q You learn logic R You read this book 5 You get into law school Argument form P a Q Q is necessary for P Q i R R is necessary for Q Il Another argument to formalize You don39t graduate unless you learn logic Moreover you learn logic only if you read this book If you don39t graduate you don39t get into law school Therefore if you get into law school you learn logic p 141 3 Assign sentence letters P You graduate Q You learn logic R You read this book 5 You get into law school Argument form P a Q Q is necessary for P Q i R R is necessary for Q P as Il Another argument to formalize You don39t graduate unless you learn logic Moreover you learn logic only if you read this book If you don39t graduate you don39t get into law school Therefore if you get into law school you learn logic p 141 3 Assign sentence letters P You graduate Q You learn logic R You read this book 5 You get into law school Argument form P a Q Q is necessary for P Q a R R is necessary for Q P as S a Q Il Checking validity Checking validity PQRS P Q Q R P S S Q Checking validity U 0 h n PgtQ QgtR Pgt iS SgtQ 39I39I I39n I39n I39n I39n I39n I39n I39n I JJJJJIJIIIJIJII 3 n 9 3 WI u on A W m H K d 0 H s d AV 0 0 AV 8 p s S Q ddddiiiiddddiiii 3 n 9 3 WI u on A W m H K d 0 H s d AV 0 0 AV 8 p s S Q IJIJIJIJIJIJIJIJiiiiiiil ddddiiiiddddiiii 3 n 9 3 WI u on A W m H K d 0 H s d AV 0 0 AV 8 p s S Q I1 IJIJIJIJIJIJIJ IJIJIJI diiddii idididi I I1 1 I1 iiiiiii IJIJIJLLLI diiddii idididi iiiiiii I1 IJIJIJLLLI d 0 H s d 0 0 8 p s S Q 3 q 9 3 WI u on A W m n K TTT TT F II II II uJI T d 0 H s d AV 0 0 AV 8 p s S Q 3 n 9 3 WI u on A W m H K TTT TT F T d 0 H s d AV 0 0 AV 8 p s S Q 3 n 9 3 WI u on A W m H K TTT TT F T d 0 H s d AV 0 0 AV 8 p s S Q 3 n 9 3 WI u on A W m H K TTT TT F T d 0 H s d AV 0 0 AV 8 p s S Q 3 n 9 3 WI u on A W m H K TTT TT F T d 0 H s d AV 0 0 AV 8 p s S Q 3 n 9 3 WI u on A W m H K TTT TT F T d O a s d ll 0 0 a T Valid p s S Q 3 n 9 3 WI u on A W m H K Preview Preview Notice 16 lines in the truth table but we only really care about 5 Preview Notice 16 lines in the truth table but we only really care about 5 We need a short cut Preview Notice 16 lines in the truth table but we only really care about 5 We need a short cut Idea Preview Notice 16 lines in the truth table but we only really care about 5 We need a short cut Idea 0 Don39t draw the whole truth table Preview Notice 16 lines in the truth table but we only really care about 5 We need a short cut Idea 0 Don39t draw the whole truth table 0 Instead think about lines in the truth table where all premises are true and conclusion is false Preview Notice 16 lines in the truth table but we only really care about 5 We need a short cut Idea 0 Don39t draw the whole truth table 0 Instead think about lines in the truth table where all premises are true and conclusion is false 0 lfwe find such a line argument form is invalid Preview Notice 16 lines in the truth table but we only really care about 5 We need a short cut Idea 0 Don39t draw the whole truth table 0 Instead think about lines in the truth table where all premises are true and conclusion is false 0 lfwe find such a line argument form is invalid o If we show that there is no such line the argument form is valid Il Math 10130 Lecture 40 28 April 2008 Exercise 1 1 5 Exercise 1 1 5 1 All Norwegians are silly 2 Every Pole is Norwegian 3 Someone hates Lenny 4 Some Pole hates Lenny 5 All Norwegians hate Lenny Il Exercise 1 1 5 1 All Norwegians are silly o VXNX 5X 2 Every Pole is Norwegian 3 Someone hates Lenny 4 Some Pole hates Lenny 5 All Norwegians hate Lenny Il Exercise 1 1 5 1 All Norwegians are silly o VXNX 5X 2 Every Pole is Norwegian o VXPX NX 3 Someone hates Lenny 4 Some Pole hates Lenny 5 All Norwegians hate Lenny Il Exercise 1 1 5 1 All Norwegians are silly o VXNX 5X 2 Every Pole is Norwegian o VXPX NX 3 Someone hates Lenny a 3X HXb 4 Some Pole hates Lenny 5 All Norwegians hate Lenny Il Exercise 1 1 5 1 All Norwegians are silly o VXNX 5X 2 Every Pole is Norwegian o VXPX NX 3 Someone hates Lenny a 3X HXb 4 Some Pole hates Lenny o 3XPXamp be 5 All Norwegians hate Lenny Il Exercise 1 1 5 1 All Norwegians are silly o VXNX 5X 2 Every Pole is Norwegian o VXPX NX 3 Someone hates Lenny a 3X HXb 4 Some Pole hates Lenny o 3XPXamp be 5 All Norwegians hate Lenny o VXNX be Il Exercise 1 6 10 Exercise 1 6 10 6 All Norwegians hate someone 7 All silly Norwegians hate a Pole 8 Some silly Pole hates all silly Norwegians 9 Some silly Pole hates all silly Norwegians who hate Lenny 10 Not every Norwegian is silly Il Exercise 1 6 10 6 All Norwegians hate someone a VXNX 3y ny 7 All silly Norwegians hate a Pole 8 Some silly Pole hates all silly Norwegians 9 Some silly Pole hates all silly Norwegians who hate Lenny 10 Not every Norwegian is silly Il Exercise 1 6 10 6 All Norwegians hate someone a VXNX 3y ny 7 All silly Norwegians hate a Pole 0 VXNXamp 5X 3yPyamp ny 8 Some silly Pole hates all silly Norwegians 9 Some silly Pole hates all silly Norwegians who hate Lenny 10 Not every Norwegian is silly Il Exercise 1 6 10 6 All Norwegians hate someone a VXNX 3y ny 7 All silly Norwegians hate a Pole 0 VXNXamp 5X 3yPyamp ny 8 Some silly Pole hates all silly Norwegians o 3XSXamp PX ampVyyamp Ny ny 9 Some silly Pole hates all silly Norwegians who hate Lenny 10 Not every Norwegian is silly Il Exercise 1 6 10 6 All Norwegians hate someone a VXNX 3y ny 7 All silly Norwegians hate a Pole 0 VXNXamp 5X 3yPyamp ny 8 Some silly Pole hates all silly Norwegians o 3XSXamp PX ampVyyamp Ny ny 9 Some silly Pole hates all silly Norwegians who hate Lenny o 3XSXamp PX ampVyyamp Ny 84 Hyb ny 10 Not every Norwegian is silly Il Exercise 1 6 10 6 All Norwegians hate someone a VXNX 3y ny 7 All silly Norwegians hate a Pole 0 VXNXamp 5X 3yPyamp ny 8 Some silly Pole hates all silly Norwegians a 3XSXamp PX ampVyyamp Ny ny 9 Some silly Pole hates all silly Norwegians who hate Lenny o 3XSXamp PX ampVyyamp Ny 84 Hyb ny 10 Not every Norwegian is silly o VXNX 5X Il Exercise 1 11 15 Exercise 1 11 15 11 Not any Norwegian is silly 12 If everyone hates Lenny then Lenny does 13 If anyone hates Lenny then Lenny does 14 Everyone hates someone but no one is hated by all 15 Someone hates everyone and some are hated by all Il Exercise 1 11 15 11 Not any Norwegian is silly o 3xNXamp 5X 12 If everyone hates Lenny then Lenny does 13 If anyone hates Lenny then Lenny does 14 Everyone hates someone but no one is hated by all 15 Someone hates everyone and some are hated by all Il Exercise 1 11 15 11 Not any Norwegian is silly o EIXNXampSX 12 If everyone hates Lenny then Lenny does a VX HXb be 13 If anyone hates Lenny then Lenny does 14 Everyone hates someone but no one is hated by all 15 Someone hates everyone and some are hated by all Il Exercise 1 11 15 11 Not any Norwegian is silly o EIXNXampSX 12 If everyone hates Lenny then Lenny does a VX HXb be 13 If anyone hates Lenny then Lenny does 0 3X HXb be 14 Everyone hates someone but no one is hated by all 15 Someone hates everyone and some are hated by all Il Exercise 1 11 15 11 Not any Norwegian is silly o EIXNXampSX 12 If everyone hates Lenny then Lenny does a VX HXb be 13 If anyone hates Lenny then Lenny does 0 3X HXb be 14 Everyone hates someone but no one is hated by all a Vx y nyamp 3XVy ny 15 Someone hates everyone and some are hated by all Il Exercise 1 11 15 11 Not any Norwegian is silly o EIXNXampSX 12 If everyone hates Lenny then Lenny does a VX HXb be 13 If anyone hates Lenny then Lenny does 0 3X HXb be 14 Everyone hates someone but no one is hated by all a Vx y nyamp 3XVy ny 15 Someone hates everyone and some are hated by all 0 HxVy ny 84 HXVy ny Il 1 FaVXFX gt GX EIX GX 1 FaVXFX gt GX EIX GX 1 Fa premise 1 FaVXFX gt GX EIX GX 1 Fa premise 2 VXFX GX premise 1 FaVXFX gt 1 Fa premise 2 VXFX GX premise 3 Fa Ga VE 2 1 FaVXFX gt 1 Fa premise 2 VXFX GX premise 3 Fa Ga VE 2 4 Ga E 13 1 FaVXFX gt 1 Fa premise 2 VXFX GX premise 3 Fa Ga VE 2 4 Ga E 13 5 3X GX 3 4 2 EIXFXVXFX gt GX EIX GX 2 EIXFXVXFX gt GX EIX GX 1 3X FX premise 2 EIXFXVXFX gt GX EIX GX 1 3X FX premise 2 VXFX GX premise 2 EIXFX VXFX gt 1 3X FX premise 2 VXFX GX premise 3 Fc SE 1 2 EIXFXVXFX gt G 1 3X FX premise 2 VXFX GX premise 3 Fc SE 1 4 Ff 66 VE 2 2 EIXFXVXFX gt GX EIX GX 3X FX VXFX GX F6 Ff 66 66 1 2 3 4 5 2 EIXFXVXFX gt GX EIX GX 3X FX VXFX GX F6 Ff 66 66 3X GX 1 2 3 4 5 6 3 VXFX gt GXVXFX gt GX VXGX ltgt FX 3 VXFX gt GXVXFX gt GX VXGX ltgt FX 1 VXFX GX premise 3 VXFX gt GXVXFX gt GX VXGX ltgt FX 1 VXFX GX premise 2 VXGX FX premise 3 VXFX gt GXVXFX gt GX VXGX ltgt FX 1 VXFX GX premise 2 VXGX FX premise 3 FX GX VE 1 3 VXFX gt GXVXFX gt GX VXGX ltgt FX 1 VXFX GX premise 2 VXGX FX premise 3 FX GX VE 1 4 GX FX VE 2 3 VXFX gt GXVXFX gt GX 1 2 3 4 5 VXFX GX VXGX FX FX GX GX FX FX ltgt GX VXGX ltgt FX premise premise VE 1 VE 2 3 VXFX gt GXVXFX gt GX 1 2 3 4 5 6 VXFX GX VXGX FX FX GX GX FX FX ltgt GX VXFX ltgt GX VXGX ltgt FX premise premise VE 1 VE 2 2H 34 VI 5 4 VXVszHXZampX y gt Hyz 4 VXVszHXZampX y gt Hyz 1 szampx y assume 4 VXVszHXZampX y gt Hyz 1 szampx y assume 2 sz ampE 1 4 VXVszHXZampX y gt Hyz 1 szampx y assume 2 sz ampE 1 3 X y ampE 1 4 szampxy sz xy Hyz szampxy Hyz assume ampE 1 ampE 1 E 32 1 4 4 1 szampx y assume 2 sz ampE 1 3 X y ampE 1 4 Hyz E 32 5 szampx y Hyz gt1 4 4 VXVszHXZampX y gt Hyz 1 szampx y assume 2 sz ampE 1 3 X y ampE 1 4 Hyz E 32 5 szampx y Hyz gt1 4 6 VzszampX y Hyz VI 5 4 VXVszHXZampX y gt Hyz 1 szampx y assume 2 sz ampE 1 3 X y ampE 1 4 Hyz E 32 5 szampx y Hyz I1394 6 VzszampX y Hyz VI 5 7 VI 6 VszszampX y Hyz 4 VXVszHXZampX y gt Hyz 1 szampx y assume 2 sz ampE 1 3 X y ampE 1 4 Hyz E 32 5 szampx y Hyz gtI1 4 6 VzszampX y Hyz VI 5 7 VszszampX y Hyz VI 6 8 VXVszszampX y Hyz VI 7 5 Elz qu 5 EIzqu qu Fz 1 Hz Fz premise 5 EIzqu qu Fz 1 Hz Fz premise 2 V2 Fz assume 5 Elz qu 1 Hz Fz premise 2 V2 Fz assume 3 F SE 1 5 Elz qu 1 Hz Fz premise 2 V2 Fz assume 3 F SE 1 4 Fc VE 2 5 EIzqu qu Fz 1 Hz Fz premise 2 V2 Fz assume 3 F SE 1 4 Fc VE 2 5 Fcamp Fc amp 34 5 Elz qu Hz Fz 1 2 3 4 5 6 premise Vz Fz assume F SE 1 Fc VE 2 Fc amp Fc amp 34 L LI 5 5 EIzqu qu Fz 1 Hz Fz premise 2 V2 Fz assume 3 F SE 1 4 Fc VE 2 5 Fc amp Fc amp 34 6 L LI 5 7 V2 Fz L 2 6 5 3392 39FZ Vz Fz 1 Hz Fz premise 2 V2 Fz assume 3 46 SE 1 4 Ff VE 2 5 Ff amp FC amp 34 6 i LI 5 7 V2 Fz L l 26 5 EIzqu qu Fz 1 Hz Fz premise 2 V2 Fz assume 3 F SE 1 4 Fc VE 2 5 Fc amp Fc amp 34 7 V2 Fz L 2 6 8 Vz Fz l 7 6 VwuFW EW FW WHH 4H H 6 VwuFW EW FW 1 VW FW premise WHH 4H H 6 VwuFW EW FW 1 VW FW premise 2 HWFW assume 6 VwuFW EW FW 1 VW FW premise 2 SW FW assume 3 Fc SE 2 6 VwuFW 1 VW FW premise 2 SW FW assume 3 Fc SE 2 4 F VE 1 6 VwuFW EW FW 1 VW FW premise 2 SW FW assume 3 Fc SE 2 4 F VE 1 5 Fcamp Fc amp 34 6 VwuFW EW FW 1 VW FW premise 2 SW FW assume 3 Fc SE 2 4 F VE 1 5 Fc amp Fc amp 34 6 L LI 5 6 VwuFW EW FW 1 VW FW premise 2 SW FW assume 3 Fc SE 2 4 F VE 1 5 Fcamp Fc amp 34 6 L LI 5 7 SW FW L 2 6 6 VW IFW EIW FW 1 VW FW premise 2 SW FW assume 3 Ff SE 2 4 46 VE 1 5 Fc amp F amp 34 6 L LI 5 7 SW FW L 2 6 6 Vw uFW EW Fw 1 VW FW premise 2 SW FW assume 3 Fc 3E 2 4 F VE 1 5 Fc amp Fc amp 34 6 L LI 5 7 SW FW L 2 6 8 aWFW l 7 7 uEX FX 7 EXFX VxuFX 1 x FX premise 7 EXFX VxuFX 1 x FX premise 2 FX assume 7 x FX Vx FX 1 ax FX premise 2 FX assume 3 3X FX 3 2 IEX FX 1 x FX premise 2 FX assume 3 3X FX 3 2 4 3X Fxamp x FX amp 31 7 EXFX VxuFX 1 x FX premise 2 FX assume 3 3X FX 3 2 4 3X Fxamp x FX amp 31 5 L Ll 4 7 EXFX VxuFX 1 x FX premise 2 FX assume 3 3X FX 3 2 4 3X Fxamp x FX amp 31 5 L Ll 4 6 Fx L 2 5 7 EXFX VxuFX 1 x FX premise 2 FX assume 3 3X FX 3 2 4 3X FXamp 3X FX amp 31 L ll 4 6 Fx L 2 5 7 EXFX VxuFX 1 x FX premise 2 FX assume 3 3X FX 3 2 4 3X FXamp 3X FX amp 31 6 Fx L I 25 7 FX l 6 7 EX FX VxuFX 1 x FX premise 2 FX assume 3 3X FX 3 2 4 3X FXamp 3X FX amp 31 5 L Ll 4 6 Fx L 2 5 7 FX l 6 8 VX FX VI 7 1 VX IFX uVX FX 1 VX IFX uVX FX Valid 1 VX IFX uVX FX Valid 1 VX FX premise 1 VX IFX uVX FX Valid 1 VX FX premise 2 VX FX assume 1 VX uFX39 Valid 1 VX FX VX FX FX premise assume VE 2 ma r lt a ma xu m AwEsmmmv xu xgt m Aw EwEv xufxgt H me xgt AU 1 VX uFX39 Valid VX FX 91me premise VX FX assume FX VE 2 FX VE 1 FXamp FX amp 34 1 VX uFX39 Valid VX FX 001me premise VX FX assume FX VE 2 FX VE 1 FXamp FX amp 34 L LI 5 premise VX FX assume FX VE 2 FX VE 1 FXamp FX amp 34 L LI 5 2 6 39 premise 2 VX FX assume 3 FX VE 2 4 FX VE 1 5 FXamp FX amp 34 6 i LI 5 7 VX FX L 39 premise 2 VX FX assume 3 FX VE 2 4 FX VE 1 5 FXamp FX amp 34 6 i LI 5 7 VX FX L l 26 8 VX FX l 7 3 qu EIX PX Fq Pq 3 qu EIX PX Fq Pq Invalid 3 qu EIX PX Fq Pq Invalid Consider the following structure 3 qu EIX PX Fq Pq Invalid Consider the following structure a Domain of discourse N 3 qu EIX PX Fq Pq Invalid Consider the following structure a Domain of discourse N 0 FX stands for X is an odd number 3 qu EIX PX Fq Pq Invalid Consider the following structure a Domain of discourse N 0 FX stands for X is an odd number 9 PX stands for X is a prime number 3 qu EIX PX Fq Pq Invalid Consider the following structure a Domain of discourse N 0 FX stands for X is an odd number 9 PX stands for X is a prime number 0 q refers to 4 3 qu EIX PX Fq Pq Invalid Consider the following structure a Domain of discourse N 0 FX stands for X is an odd number 9 PX stands for X is a prime number 0 q refers to 4 Fq V 3X PX says 4 is odd or there is a prime number 3 qu EIX PX Fq Pq Invalid Consider the following structure a Domain of discourse N 0 FX stands for X is an odd number 9 PX stands for X is a prime number 0 q refers to 4 Fq V 3X PX says 4 is odd or there is a prime number True 3 qu EIX PX Fq Pq Invalid Consider the following structure a Domain of discourse N 0 FX stands for X is an odd number 9 PX stands for X is a prime number 0 q refers to 4 Fq V 3X PX says 4 is odd or there is a prime number True Fq V Pq says 4 is odd or 4 a prime number Il 3 qu EIX PX Fq Pq Invalid Consider the following structure a Domain of discourse N 0 FX stands for X is an odd number a PX stands for X is a prime number 0 q refers to 4 Fq V 3X PX says 4 is odd or there is a prime number True Fq V Pq says 4 is odd or 4 a prime number False Il 5 VX PX VX FX VX PX FX 5 VX PX VX FX VX PX FX Valid Aw EwEv xu xgt gt Xi xgt H Umgt OE gtXQQXgt x xgtgtxl xgt Amv 5 VX PX VX FX VX PX FX Valid 1 VX PX V VX FX premise 2 VX PX assume a E m AwEsmmmv XQ xgt m Aw EwEv xu xgtgt XQ xgt H Umgt OE gt XQQ xgt XL xgt gt XQ xgt Amv 9 gt a m gt a x AwEsmmmv Xi XQ m Aw EwEv gt m xu xgt gt XQ xgt H Umgt OE gt XQQ xgt XL xgt gt XQ xgt Amv 6 E 3 gt xxgt m Am 2 xu gt lt a E m AwEsmmmv XQ xgt m Aw EwEv xu xgt gt Xi xgt H Umgt OE gt XQQ xgt XL xgt gt XQ xgt Amv 3m WV 6 E Am 2 a ma AwEsmmmv Aw EwEv cm gt xxgt xgt 0 cm gtxxgt m Va 1 e m xgt m xgtgt xgt H Umgt OE gt XQQ xgt XL xgt gt XQ xgt Amv 3m 7i 3 gt g xgt xgt o 6 E 3 gt g xgt m Am 2 xu gt lt a E m A m A wEsmmmv Xi xgt w EwEv xu xgt gt Xi xgt H Umgt OE gt XQQ xgt XL xgt gt XQ xgt Amv AwEsmmmv xu xgt N 3m 7i cmgtxxgt xm xgt 6 E 3 gt x3 xgt Mm E xu gt A A N Na wEsmmmv Xi xgt w EwEv xu xgt gt XQ xgt H aim n Umgt OE gt XQQ xgt XL xgt gt XQ xgt Amv C E xu m AwEsmmmv xu xgt N 3m 7i 3 gt A xgt x1 xgt o 6 E 3 gt x3 xgt Mm E xu gt A A m E wEsmmmv Xi xgt w EwEv xu xgt gt Xi xgt H aim n Umgt on gt x5 xgt xgt gt xgt E w E xu gt Q A 6 xu wEsmmmv xu xgt E 7i Axugtxxgt xgt A A A A M0 E Axu gt v xgt A A A m E wEsmmmv Xi xgt w EwEv xu xgt gt Xi xgt H m m E xu gt A m m Umgt Ao E Axu gt g xgt 3 Am 2 xu gt 0 AA ma xu m AwEsmmmv xu xgt N AWN 7i Axugtxxgt xm xgt me E Axu gt g xgt A A A m gtV xu gt xn N Na wEsmmmv Xi xgt w EwEv xu xgt gt Xi xgt H aim n Umgt OE gt XQQ xgt XL xgt gt XQ xgt Amv A0 1 Axugtxxgt xu V3 Ao E Axu gt x3 xgt 3 Am 2 xu gt 0 AA ma xu m AwEsmmmv xu xgt N AWN 7i Axugtxxgt xm xgt A A A A A o E Axu gt g xgt m gtV xu gt xn N Na wEsmmmv Xi xgt w EwEv xu xgt gt Xi xgt H aim n Umgt OE gt XQQ xgt XL xgt gt XQ xgt Amv Ao v Axugtxnvxgt xu 5 Am E Axu gt 3 xgt 3 Am 2 xu gt 0 AA ma xu m AwEsmmmv xu xgt N AWN v C0 gt xmvxgt xn xgt o A A A A A o E Axu gt g xgt m m gtV xu gt xn e N Na m wEsmmmv XnA xgt m w EwEv xuxgtgtxmxgt H Umgt OE gt XQQ xgt XL xgt gt XQ xgt Amv 3 ma Axugtxxgt S o l Axugtxxgt xu a E Axu gt 3 xgt 3 m gtV xu gt xn o A m3 xu m wEsmmmv xu xgt N mm 7i Axugtxxgt xm xgt o o E Axu gt g xgt m gtV xu gt xn N Na wEsmmmv Xi xgt w EwEv xu xgt gt Xi xgt H m w m m A A A A A A A A A A A A Umgt OE gt XQQ xgt XL xgt gt XQ xgt Amv 7 VX PX gt VX FX VXPX gt FX 7 VX PX gt VX FX VXPX gt FX Invalid 7 VX PX gt VX FX VXPX gt FX Invalid Consider the following structure 7 VX PX gt VX FX VXPX gt FX Invalid Consider the following structure 0 Domain of discourse N 7 VX PX gt VX FX VXPX gt FX Invalid Consider the following structure 0 Domain of discourse N o PX stands for X is an even number 7 VX PX gt VX FX VXPX gt FX Invalid Consider the following structure 0 Domain of discourse N o PX stands for X is an even number 0 FX stands for X is a odd number 7 VX PX gt VX FX VXPX gt FX Invalid Consider the following structure 0 Domain of discourse N o PX stands for X is an even number 0 FX stands for X is a odd number 7 VX PX gt VX FX VXPX gt FX Invalid Consider the following structure 0 Domain of discourse N o PX stands for X is an even number 0 FX stands for X is a odd number VX PX VX FX says If every number is even then every number is odd 7 VX PX gt VX FX VXPX gt FX Invalid Consider the following structure 0 Domain of discourse N o PX stands for X is an even number 0 FX stands for X is a odd number VX PX VX FX says If every number is even then every number is odd True 7 VX PX gt VX FX VXPX gt FX Invalid Consider the following structure 0 Domain of discourse N o PX stands for X is an even number 0 FX stands for X is a odd number VX PX VX FX says If every number is even then every number is odd True VXPX FX says For every number X if X is even then X is odd Il 7 VX PX gt VX FX VXPX gt FX Invalid Consider the following structure 9 Domain of discourse N o PX stands for X is an even number 0 FX stands for X is a odd number VX PX VX FX says If every number is even then every number is odd True VXPX FX says For every number X if X is even then X is odd False Il 10 VX PX ltgt FX EIX IFX VX IPX 10 VX PX ltgt Invalid 10 VX PX ltgt FX EIX IFX VX IPX Invalid Consider the following structure 10 VX PX ltgt FX EIX IFX VX IPX Invalid Consider the following structure 0 Domain of discourse humans 10 VX PX ltgt FX EIX IFX VX IPX Invalid Consider the following structure 0 Domain of discourse humans 0 PX stands for X is an unmarried male 10 VX PX ltgt FX EIX IFX VX IPX Invalid Consider the following structure 0 Domain of discourse humans 0 PX stands for X is an unmarried male 0 FX stands for X is a bachelor 10 VX PX ltgt FX EIX IFX VX IPX Invalid Consider the following structure 0 Domain of discourse humans 0 PX stands for X is an unmarried male 0 FX stands for X is a bachelor 10 VX PX ltgt FX EIX IFX VX IPX Invalid Consider the following structure 0 Domain of discourse humans 0 PX stands for X is an unmarried male 0 FX stands for X is a bachelor VXPX ltgt FX says For every human X X is an unmarried male if and only ifX is a bachelor 10 VX PX ltgt FX EIX IFX VX IPX Invalid Consider the following structure 0 Domain of discourse humans 0 PX stands for X is an unmarried male 0 FX stands for X is a bachelor VXPX ltgt FX says For every human X X is an unmarried male if and only ifX is a bachelor True 10 VX PX ltgt FX EIX IFX VX IPX Invalid Consider the following structure 0 Domain of discourse humans 0 PX stands for X is an unmarried male 0 FX stands for X is a bachelor VXPX ltgt FX says For every human X X is an unmarried male if and only ifX is a bachelor True HX iFX says There is some non bachelor 10 VX PX ltgt FX EIX IFX VX IPX Invalid Consider the following structure 0 Domain of discourse humans 0 PX stands for X is an unmarried male 0 FX stands for X is a bachelor VXPX ltgt FX says For every human X X is an unmarried male if and only ifX is a bachelor True HX iFX says There is some non bachelor True 10 VX PX ltgt FX EIX IFX VX IPX Invalid Consider the following structure 0 Domain of discourse humans 0 PX stands for X is an unmarried male 0 FX stands for X is a bachelor VXPX ltgt FX says For every human X X is an unmarried male if and only ifX is a bachelor True HX iFX says There is some non bachelor True VX iPX says All humans are non unmarried bachelors Il 10 VX PX ltgt FX EIX IFX VX IPX Invalid Consider the following structure 0 Domain of discourse humans 0 PX stands for X is an unmarried male 0 FX stands for X is a bachelor VXPX ltgt FX says For every human X X is an unmarried male if and only ifX is a bachelor True HX iFX says There is some non bachelor True VX iPX says All humans are non unmarried bachelors False Il

### BOOM! Enjoy Your Free Notes!

We've added these Notes to your profile, click here to view them now.

### You're already Subscribed!

Looks like you've already subscribed to StudySoup, you won't need to purchase another subscription to get this material. To access this material simply click 'View Full Document'

## Why people love StudySoup

#### "I was shooting for a perfect 4.0 GPA this semester. Having StudySoup as a study aid was critical to helping me achieve my goal...and I nailed it!"

#### "I made $350 in just two days after posting my first study guide."

#### "Knowing I can count on the Elite Notetaker in my class allows me to focus on what the professor is saying instead of just scribbling notes the whole time and falling behind."

#### "It's a great way for students to improve their educational experience and it seemed like a product that everybody wants, so all the people participating are winning."

### Refund Policy

#### STUDYSOUP CANCELLATION POLICY

All subscriptions to StudySoup are paid in full at the time of subscribing. To change your credit card information or to cancel your subscription, go to "Edit Settings". All credit card information will be available there. If you should decide to cancel your subscription, it will continue to be valid until the next payment period, as all payments for the current period were made in advance. For special circumstances, please email support@studysoup.com

#### STUDYSOUP REFUND POLICY

StudySoup has more than 1 million course-specific study resources to help students study smarter. If you’re having trouble finding what you’re looking for, our customer support team can help you find what you need! Feel free to contact them here: support@studysoup.com

Recurring Subscriptions: If you have canceled your recurring subscription on the day of renewal and have not downloaded any documents, you may request a refund by submitting an email to support@studysoup.com

Satisfaction Guarantee: If you’re not satisfied with your subscription, you can contact us for further help. Contact must be made within 3 business days of your subscription purchase and your refund request will be subject for review.

Please Note: Refunds can never be provided more than 30 days after the initial purchase date regardless of your activity on the site.