Solutions (or Hints) to Chapter-end Exercises ~ 377
1.12 P /\ (P ==> -, Q) == P /\ (-, P V -, Q) == (P /\ -, P) V (P /\ -, Q) ==
F v (P /\ -, Q) == P /\ -, Q
So (P /\ (P ==> -, Q)) v (Q ==> -, Q) == (P /\ -, Q) V (-, Q v -, Q) ==
(P A -, Q) v -, Q == -, Q. Hence
((P /\ (P ==> -, Q)) v (Q ==> -, Q)) ==> -, Q == (--, Q ==> -, Q) == -, (--, Q)
v-,Q==Qv-,Q=T
1.13 ex == (Q /\ -,R /\ -,5) v (R /\ 5)
== (Q /\ -, R /\ -, 5) v (R /\ 5 /\ Q) v (R /\ S /\ -, Q)
== (Q 1\ -,R 1\ -,5) v (Q /\ R /\ 5) v (-,Q 1\ R 1\ S)
1.14 Let the literals be P, Q, R. Then
ex == 110 v 100 v 010 v 000
== ((P 1\ Q /\ -, R) v (P 1\ -, Q 1\ -, R)) v (010 v 000)
== (P 1\ -, R) v (-, P 1\ Q 1\ -, R) v (-, P 1\ -, Q /\ -, R))
== (P 1\ -, R) v (--, P 1\ -, R)
== -, R
1.15 (a) The given premises are: (i) P ==> Q, (ii) R ==> -, Q. To derive
P ==> -, R, we assume (iii) P as an additional premise and deduce -, R.
1. P
Premise (iii)
2. P ==> Q
Premise (i)
3. Q
RI..
4. -, (--, Q)
1 7
5. R ==> -, Q
Premise (ii)
6. -, R
RI s
7. P ==> -, R
Lines 1 and 6
Hence the argument is valid.
(b) Valid
(c) Let the given premises be (i) P, (ii) Q, (iii) -, Q ==> R,
(iv) Q ==> -, R. Then
1. Q
Premise (ii)
2. -, R
Premise (iv)
Hence the given argument is valid.
(d) Let the given premises be (i) 5, (ii) P, (iii) P ==> Q 1\ R,
(iv) Q v S ==> T. Then
1. P
Premise (ii)
2. Q /\ R
Premise (iii)
3. Q
RI 3
4. Q v 5
Rlj
5. T
Premise (iv)
Hence the argument is valid. Note that in (c) and (d) the conclusions
are obtained without using some of the given premises.