80-211 Spring 2003
Answer key
Note: These are the answers to some of the
questions in the book and some answers to problems not in the book, most of
these were not problems you had to do for homework, but are similar.
Problems from pages 27-28:
1-a P |- Qà (P & Q)
1 (1) P A
2 (2) Q A
1,2 (3) P & Q 1,2 &-I
1 (4) Qà (P & Q) 2,3 CP
1-f (Pà R) & (Qà R) |- (P v Q)à R
1 (1) (Pà R) & (Qà R) A
2 (2) P v Q A
3 (3) P A
1 (4) Pà R 1, &-E
1,3 (5) R 3,4 MPP
6 (6) Q A
1 (7) Qà R 1, &-E
1, 6 (8) R 6,7 MPP
1,2 (9) R 2, 3, 5, 6, 8 v-E
1 (10) (P v Q)à R 2, 9 CP
1-g Pà Q, Rà S |- (P & R) à (Q & S)
1 (1) Pà Q A
2 (2) Rà S A
3 (3) P & R A
3 (4) P 3, &-E
1, 3 (5) Q 1, 4 MPP
3 (6) R 3, &-E
2, 3 (7) S 2, 6 MPP
1, 2, 3 (8) Q & S 5, 7 &-I
1, 2 (9) (P & R) à (Q & S) 3, 8 &-I
1-i Pà (Q & R) |- (Pà Q) & (Pà R)
1 (1) Pà (Q & R) A
2 (2) P A
1, 2 (3) Q & R 1,2 MPP
1, 2 (4) Q 3, &-E
1 (5) Pà Q 2, 4 CP
1, 2 (6) R 3, &-E
1 (7) Pà R 2, 6 CP
1 (8) (Pà Q) & (Pà R) 5, 7 &-I
Problems on page 33:
1-c P « Q |- ~ P « ~Q
1 (1) P « Q A
1 (2) (Pà Q) & (Qà
P) 1, def «
1 (3) Pà Q 2, &-E
1 (4) Qà P 2, &-E
5 (5) ~Q A
1, 5 (6) ~P 3, 5 MTT
1 (7) ~Qà ~P 5,6 CP
8 (8) ~P A
1, 8 (9) ~Q 4, 8 MTT
1 (10) ~Pà ~Q 8, 9 CP
1 (11) (~Pà ~Q) & (~Qà ~P) 7, 10 &-I
1 (12) ~P « ~Q 11,
def «
1-e (P v Q) « P |- Qà
P
1 (1) (P v Q) « P A
1 (2) ((P v Q)à P) & (Pà
(P v Q)) 1, def «
3 (3) Q A
3 (4) P v Q 3, v-I
1 (5) (P v Q)à P 2, &-E
1, 3 (6) P 4,5 MPP
1 (7) Qà P 3, 6 CP
Problems on page 41:
1-c P & (Q v R) -||- (P & Q) v (P & R)
Part a: P & (Q v R) |- (P & Q) v (P & R)
1 (1) P & (Q v R) A
1 (2) P 1, &-E
1 (3) Q v R 1, &-E
4 (4) Q A
1, 4 (5) P & Q 2, 4 &-I
1, 4 (6) (P & Q) v (P & R) 5, v-I
7 (7) R A
1, 7 (8) P & R 2, 7 &-I
1, 7 (9) (P & Q) v (P & R) 8, v-I
1 (10) (P & Q) v (P & R) 3, 4, 6, 7, 9 v-E
Part b: (P & Q) v (P & R) |- P & (Q v R)
1 (1) (P & Q) v (P & R) A
2 (2) P & Q A
2 (3) P 2, &-E
2 (4) Q 2, &-E
2 (5) Q v R 4, v-I
2 (6) P & (Q v R) 3, 5 &-I
7 (7) P & R A
7 (8) P 7, &-E
7 (9) R 7, &-E
7 (10) Q v R 9, v-I
7 (11) P & (Q v R) 8, 10 &-I
1 (12) P & (Q v R) 1, 2, 6, 7, 11, v-E
1-f ~ (P v Q) -||- ~P & ~Q
Part a: ~ (P v Q) |- ~P & ~Q
1 (1) ~ (P v Q) A
2 (2) P A
2 (3) P v Q 2, v-I
1, 2 (4) ~ (P v Q) & (P v Q) 1, 3 &-I
1 (5) ~P 2, 4 RAA
6 (6) Q A
6 (7) P v Q 6, v-I
1, 6 (8) ~ (P v Q) & (P v Q) 1, 7 &-I
1 (9) ~Q 6, 8 RAA
1 (10) ~P & ~Q 5, 9 &-I
Part b: ~P & ~Q |- ~ (P v Q)
1 (1) ~P & ~Q A
2 (2) P v Q A
3 (3) P A
1 (4) ~P 1, &-E
1, 3 (5) P & ~P 3, 4 &-I
3 (6) ~ (~P & ~Q) 1, 5 RAA
7 (7) Q A
1 (8) ~Q 1, &-E
1, 7 (9) Q & ~Q 7, 8 &-I
7 (10) ~ (~P & ~Q) 1, 9 RAA
1, 2 (11) ~ (~P & ~Q) 2, 3, 6, 7, 10, v-E
1, 2 (12) ~ (~P & ~Q) & (~P & ~Q) 1, 11 &-I
1 (13) ~ (P v Q)
Note: The following 4
proofs use derived rules.
|- (P®Q) v (Q®P)
(1) P
v ~P TI
44
2 (2) P A
2 (3) Q®P 2
SI 50
2 (4) (P®Q)
v (Q®P) 3,
v-I
5 (5) ~P A
5 (6) P®Q 5
SI 51
5 (7) (P®Q)
v (Q®P) 6,
v-I
(8) (P®Q)
v (Q®P) 1,
2, 4, 5, 7 v-E
~(P v Q) v ~(~P v ~Q) |- P«Q
1 (1) ~(P
v Q) v ~(~P v ~Q) A
2 (2) ~(P
v Q) A
2 (3) ~P
& ~ Q 2
SI DeMorgan
2 (4) ~P 3,
&-E
2 (5) ~Q 3,
&-E
2 (6) P®Q 4
SI 51
2 (7) Q®P 5
SI(S) 51
2 (8) (P®Q)
& (Q®P) 6, 7 &-I
9 (9) ~(~P
v ~Q) A
9 (10) P
& Q 9
SI DeMorgan (prob 1-h)
9 (11) P 10
&-E
9 (12) Q 10
&-E
9 (13) Q®P 11
SI-50
9 (14) P®Q 12
SI(S)-50
9 (15) (P®Q)
& (Q®P) 13, 14 &-I
1 (16) (P®Q)
& (Q®P) 1, 2, 8, 9, 15 v-E
1 (17) P«Q 16
« Def
P v Q, (Q®R) & (~P v S), (Q & R)®T |- T v S
1 (1) P
v Q A
2 (2) (Q®R) & (~P v S) A
3 (3) (Q
& R)®T A
4 (4) P A
2 (5) ~P
v S 2,
&-E
4 (6) ~~P 4, DN
2,
4 (7) S 5,
6 MTP (aka disjunctive syllogism)
2,
4 (8) T v S 7,
v-I
9 (9) Q A
2 (10) Q®R 2,
&-E
2,
9 (11) R 9,
10 MPP
2,
9 (12) Q & R 9,
11 &-I
2,
3, 9 (13) T 3,
12 MPP
2,
3, 9 (14) T v S 13,
v-I
1,
2, 3 (15) T v S 1,
4, 8, 9, 14 v-E