ANSWERS TO EXERCISES IN VOLUME I, CHAPTER 6 6-1. a) 1 |Kv-I P +--------------------- 2 | |-K&I A | +--------------------- 3 | |Kv-I 1,R 4 | |-K 2,&E 5 | |-I 3,4,vE 6 | |I 2,&E 7 |-(-K&I) 2-6,-I 6-1. b) 1 |-C>A P 2 |B>-A P 3 |B P +--------------------- 4 |-A 2,3,>E 5 | |-C A | +--------------------- 6 | |-C>A 1,R 7 | |A 5,6,>E 8 | |-A 4,R 9 |--C 5-8,-I 10 |C 9,-E 6-1. c) 1 |-D>-K P 2 |K P 3 |-KvH P +--------------------- 4 |H 2,3,vE 5 | |-D A | +--------------------- 6 | |-D>-K 1,R 7 | |-K 5,6,>E 8 | |K 2,R 9 |--D 5-8,-I 10 |D 9,-E 11 |H & D 4,10,&I 6-1. d) 1 |-F>G P 2 |G>-E P +--------------------- 3 | |E A | +--------------------- 4 | | |G A | | +-------------------- 5 | | |G>-E 2,R 6 | | |-E 4,5,>E 7 | | |E 3,R 8 | |-G 4-7,-I 9 | | |-F A | | +-------------------- 10 | | |-F>G 1,R 11 | | |G 9,10,>E 12 | | |-G 8,R 13 | |--F 9-12,-I 14 | |F 13,-E 15 |E>F 3-14,>I 6-1. e) 1 |A<->-B P +--------------------- 2 | |-A A | +--------------------- 3 | |A<->-B 1,R 4 | |-B>A 3,<>E 5 | | |-B A | | +-------------------- 6 | | |-B>A 4,R 7 | | |A 5,6,>E 8 | | |-A 2,R 9 | |--B 5-8,-I 10 | |B 9,-E 11 |-A>B 2-10,>I 6-1. f) 1 |AvB P 2 |B>C P 3 |-CvD P +--------------------- 4 | |-D A | +--------------------- 5 | |-CvD 3,R 6 | |-C 4,5,vE 7 | | |B A | | +-------------------- 8 | | |B>C 2,R 9 | | |C 7,8,>E 10 | | |-C 6,R 11 | |-B 7-10,-I 12 | |AvB 1,R 13 | |A 11,12,vE 14 |-D>A 4-13,>I 6-2. a) 1 |(AvB)&((AvB)>C) P +--------------------- 2 |AvB 1,&E 3 |(AvB)>C 1,&E 4 |C 2,3,>E 6-2. b) 1 |A P 2 |(AvB)<->((A>K)&(B>K)) P +--------------------- 3 |(AvB)>((A>K)&(B>K)) 2,<>E 4 |AvB 1,vI 5 |(A>K)&(B>K) 3,4,>E 6 |A>K 5,&E 7 |K 1,6,>E 6-2. c) 1 |[A>(Dv-B)]&{[A>(Dv-B)]>(B>A)} P +--------------------- 2 |A>(Dv-B) 1,&E 3 |(A>(Dv-B))>(B>A) 1,&E 4 |B>A 2,3,>E 6-3. a) 1 |R P +--------------------- 2 |RvD 1,vI 3 |RvK 1,vI 4 |(RvD)&(RvK) 2,3,&I 6-3. b) 1 |(-A&B)vC P 2 |AvD P +--------------------- 3 | |-C A | +--------------------- 4 | |(-A&B)vC 1,R 5 | |-A&B 3,4,vE 6 | |-A 5,&E 7 | |AvD 2,R 8 | |D 6,7,vE 9 |-C>D 3-8,>I 6-3. c) 1 |-(Hv-D) P 2 |F>H P +--------------------- 3 | |F A | +--------------------- 4 | |F>H 2,R 5 | |H 3,4,>E 6 | |Hv-D 5,vI 7 | |-(Hv-D) 1,R 8 |-F 3-7,-I 6-3. d) 1 |F>(O>M) P +--------------------- 2 | |F&O A | +--------------------- 3 | |F 2,&E 4 | |F>(O>M) 1,R 5 | |O>M 3,4,>E 6 | |O 2,&E 7 | |M 5,6,>E 8 |(F&O)>M 2-7,>I 6-3. e) 1 |P&-Q P +--------------------- 2 | |-R A | +--------------------- 3 | | |P>(RvQ) A | | +-------------------- 4 | | |P&-Q 1,R 5 | | |P 4,&E 6 | | |RvQ 3,5,>E 7 | | |-R 2,R 8 | | |Q 6,7,vE 9 | | |-Q 4,&E 10 | |-(P>(RvQ)) 3-9,-I 11 |-R>-(P>(RvQ)) 2-10,>I 6-3. f) 1 |(K&G)>S P +--------------------- 2 | |K A | +--------------------- 3 | | |G A | | +-------------------- 4 | | |(K&G)>S 1,R 5 | | |K 2,R 6 | | |K&G 3,5,&I 7 | | |S 4,6,>E 8 | |G>S 3-7,>I 9 |K>(G>S) 2-8,>I 6-3. g) 1 |-(A&-F) P 2 |D>A P +--------------------- 3 | |D A | +--------------------- 4 | | |-F A | | +-------------------- 5 | | |D>A 2,R 6 | | |D 3,R 7 | | |A 5,6,>E 8 | | |A&-F 4,7,&I 9 | | |-(A&-F) 1,R 10 | |--F 4-9,-I 11 | |F 10,-E 12 |D>F 3-11,>I 6-3. h) 1 |-(N>I) P 2 |-I>C P +--------------------- 3 | |I A | +--------------------- 4 | | |N A | | +-------------------- 5 | | |I 3,R 6 | |N>I 4-5,>I 7 | |-(N>I) 1,R 8 |-I 3-7,-I 9 |C 2,8,>E 6-3. i) 1 |-M>-L P 2 |-L>-K P +--------------------- 3 | |K A | +--------------------- 4 | | |-M A | | +-------------------- 5 | | |-M>-L 1,R 6 | | |-L 4,5,>E 7 | | |-L>-K 2,R 8 | | |-K 6,7,>E 9 | | |K 3,R 10 | |--M 4-9,-I 11 | |M 10,-E 12 |K>M 3-11,>I 6-3. j) 1 |QvF P 2 |Q>A P 3 |F>A P +--------------------- 4 | |-A A | +--------------------- 5 | | |Q A | | +-------------------- 6 | | |Q>A 2,R 7 | | |A 5,6,>E 8 | | |-A 4,R 9 | |-Q 5-8,-I 10 | |QvF 1,R 11 | |F 9,10,vE 12 | |F>A 3,R 13 | |A 11,12,>E 14 |--A 4-13,-I 15 |A 14,-E 6-3. k) 1 |-(S&T) P 2 |SvT P +--------------------- 3 | |-S A | +--------------------- 4 | |SvT 2,R 5 | |T 3,4,vE 6 |-S>T 3-5,>I 7 | |T A | +--------------------- 8 | | |S A | | +-------------------- 9 | | |T 7,R 10 | | |S&T 8,9,&I 11 | | |-(S&T) 1,R 12 | |-S 8-11,-I 13 |T>-S 7-12,>I 14 |-S<->T 6,13,<>I 6-3. l) 1 |-C>(AvB) P 2 |-D>(Cv-B) P 3 |-(CvD) P +--------------------- 4 | |C A | +--------------------- 5 | |CvD 4,vI 6 | |-(CvD) 3,R 7 |-C 4-6,-I 8 |AvB 1,7,>E 9 | |D A | +--------------------- 10 | |CvD 9,vI 11 | |-(CvD) 3,R 12 |-D 9-11,-I 13 |Cv-B 2,12,>E 14 |(AvB)&(Cv-B) 8,13,&I 6-3. m) 1 |G<->-H P +--------------------- 2 |G>-H 1,<>E 3 |-H>G 1,<>E 4 | |-G A | +--------------------- 5 | | |-H A | | +-------------------- 6 | | |-H>G 3,R 7 | | |G 5,6,>E 8 | | |-G 4,R 9 | |--H 5-8,-I 10 | |H 9,-E 11 |-G>H 4-10,>I 12 | |H A | +--------------------- 13 | | |G A | | +-------------------- 14 | | |G>-H 2,R 15 | | |-H 13,14,>E 16 | | |H 12,R 17 | |-G 13-16,-I 18 |H>-G 12-17,>I 19 |-G<->H 11,18,<>I 6-3. n) 1 |P<->Q P +--------------------- 2 | |P<->-Q A | +--------------------- 3 | | |P A | | +-------------------- 4 | | |P<->Q 1,R 5 | | |P>Q 4,<>E 6 | | |Q 3,5,>E 7 | | |P<->-Q 2,R 8 | | |P>-Q 7,<>E 9 | | |-Q 3,8,>E 10 | |-P 3-9,-I 11 | |-Q>P 2,<>E 12 | | |-Q A | | +-------------------- 13 | | |-Q>P 11,R 14 | | |P 12,13,>E 15 | | |-P 10,R 16 | |--Q 12-15,-I 17 | |Q 16,-E 18 | |P<->Q 1,R 19 | |Q>P 18,<>E 20 | |P 17,19,>E 21 |-(P<->-Q) 2-20,-I 6-3. o) 1 |(N>S)&(G>D) P 2 |(SvD)>{[F>(FvK)]>(N&G)} P +--------------------- 3 |N>S 1,&E 4 |G>D 1,&E 5 | |N A | +------------- 6 | |N>S 3,R 7 | |S 5,6,>E 8 | |SvD 7,vI 9 | |(SvD)>((F>(FvK))>(N&G)) 2,R 10 | |(F>(FvK))>(N&G) 8,9,>E 11 | | |F A | | +------------ 12 | | |FvK 11,vI 13 | |F>(FvK) 11-12,>I 14 | |N&G 10,13,>E 15 | |G 14,&E 16 |N>G 5-15,>I 17 | |G A | +--------------- 18 | |G>D 4,R 19 | |D 17,18,>E 20 | |SvD 19,vI 21 | |(SvD)>((F>(FvK))>(N&G)) 2,R 22 | |(F>(FvK))>(N&G) 20,21,>E 23 | | |F A | | +-------------- 24 | | |FvK 23,vI 25 | |F>(FvK) 23-24,>I 26 | |N&G 22,25,>E 27 | |N 26,&E 28 |G>N 17-27,>I 29 |N<->G 16,28,<>I 6-3. p) 1 |(S&B)>K P 2 |(G>P)&(GvP) P 3 |-B<->(-P&G) P +--------------------- 4 | |S A | +--------------------- 5 | | |-B A | | +-------------------- 6 | | |-B<->(-P&G) 3,R 7 | | |-B>(-P&G) 6,<>E 8 | | |-P&G 5,7,>E 9 | | |-P 8,&E 10 | | |G 8,&E 11 | | |(G>P)&(GvP) 2,R 12 | | |G>P 11,&E 13 | | |P 10,12,>E 14 | |--B 5-13,-I 15 | |B 14,-E 16 | |S&B 4,15,&I 17 | |(S&B)>K 1,R 18 | |K 16,17,>E 19 |S>K 4-18,>I 6-3. q) 1 |CvB P 2 |-(C&-B) P 3 |-(-C&B) P +--------------------- 4 | |-C A | +--------------------- 5 | |CvB 1,R 6 | |B 4,5,vE 7 | |-C&B 4,6,&I 8 | |-(-C&B) 3,R 9 |--C 4-8,-I 10 |C 9,-E 11 | |-(Bv-C) A | +--------------------- 12 | | |B A | | +-------------------- 13 | | |Bv-C 12,vI 14 | | |-(Bv-C) 11,R 15 | |-B 12-14,-I 16 | | |-C A | | +-------------------- 17 | | |Bv-C 16,vI 18 | | |-(Bv-C) 11,R 19 | |--C 16-18,-I 20 | |C 19,-E 21 | |C&-B 15,20,&I 22 | |-(C&-B) 2,R 23 |--(Bv-C) 11-22,-I 24 |Bv-C 23,-E 25 |C&(Bv-C) 10,24,&I 6-3. r) 1 |H>(D>K) P 2 |(K&M)>P P 3 |I>-(M>P) P +--------------------- 4 | |H A | +--------------------- 5 | | |D A | | +-------------------- 6 | | | |I A | | | +------------------ 7 | | | |I>-(M>P) 3,R 8 | | | |-(M>P) 6,7,>E 9 | | | |H 4,R 10 | | | |D 5,R 11 | | | |H>(D>K) 1,R 12 | | | |D>K 9,11,>E 13 | | | |K 10,12,>E 14 | | | | |M A | | | | +---------------- 15 | | | | |K 13,R 16 | | | | |K&M 14,15,&I 17 | | | | |(K&M)>P 2,R 18 | | | | |P 16,17,>E 19 | | | |M>P 14-18,>I 20 | | |-I 6-19,-I 21 | |D>-I 5-20,>I 22 |H>(D>-I) 4-21,>I 6-4. -X X|Y X|Y -Y -X |E -Y |E X|Y |I