Solutions to Chapter 19 Additional rules for TFL

A. The following proofs are missing their citations (rule and line numbers). Add them wherever they are required:

B. Give a proof for each of these arguments:

1. 1.
   ​

   $E\vee F$, $F\vee G$, $\neg F\therefore E\wedge G$

   Line number	Subproof level	Formula	Justification
   1	0	$E\vee F$
   2	0	$F\vee G$
   3	0	$\neg F$
   4	0	$E$	DS 1, 3
   5	0	$G$	DS 2, 3
   6	0	$E\wedge G$	$\wedge$I 4, 5

2. 2.
   ​

   $M\vee (N\to M)\therefore \neg M\to \neg N$

   Line number	Subproof level	Formula	Justification
   1	0	$M\vee (N\to M)$
   2	open subproof, 1	$\neg M$
   3	1	$N\to M$	DS 1, 2
   4	1	$\neg N$	MT 3, 2
   5	close subproof, 0	$\neg M\to \neg N$	$\to$I 2–4

3. 3.
   ​

   $(M\vee N)\wedge (O\vee P)$, $N\to P$, $\neg P\therefore M\wedge O$

   Line number	Subproof level	Formula	Justification
   1	0	$(M\vee N)\wedge (O\vee P)$
   2	0	$N\to P$
   3	0	$\neg P$
   4	0	$\neg N$	MT 2, 3
   5	0	$M\vee N$	$\wedge$E 1
   6	0	$M$	DS 5, 4
   7	0	$O\vee P$	$\wedge$E 1
   8	0	$O$	DS 7, 3
   9	0	$M\wedge O$	$\wedge$I 6, 8

4. 4.
   ​

   $(X\wedge Y)\vee (X\wedge Z)$, $\neg (X\wedge D)$, $D\vee M$ $\therefore$$M$

   Line number	Subproof level	Formula	Justification
   1	0	$(X\wedge Y)\vee (X\wedge Z)$
   2	0	$\neg (X\wedge D)$
   3	0	$D\vee M$
   4	open subproof, 1	$X\wedge Y$
   5	1	$X$	$\wedge$E 4
   6	close subproof, open subproof, 1	$X\wedge Z$
   7	1	$X$	$\wedge$E 6
   8	close subproof, 0	$X$	$\vee$E 1, 4–5, 6–7
   9	open subproof, 1	$D$
   10	1	$X\wedge D$	$\wedge$I 8, 9
   11	1	$\perp$	$\neg$E 10, 2
   12	close subproof, 0	$\neg D$	$\neg$I 9–11
   13	0	$M$	DS 3, 12