1.

2. Simplification
3. Modus ponens
4. Modus tollens
5. Disjunctive syllogism

2.

1. The argument is translated as follows:
T S
T C
S
----------------------------------------------------
C

The inference rules used are:
From ( T S )   and S
by modus tollens we deduce T
From T and ( T C )
by disjunctive syllogism we conclude C.

2. The argument is translated as follows:
( S W )     ( R C )
-----------------------------------------------------
( W C )

The inference rules used are:
( S W ) is equivalent to ( S W )
by implication and De Morgan
Also ( R C ) is equivalent to ( R C ) by De Morgan.
Hence ( S W ) ( R C ) is equivalent to ( S W ) ( R C ), which is equivalent to
( S R ) ( W C )
Hence by simplification ( W C )

3. The argument is translated as follows:
C S
O L
O C
-----------------------------------------------------
S L

The inference rules used are:
S C
by Contrapositive of C S
C O
by Contrapositive of O C
By hypothetical syllogism from the last two
S O
By another hypothetical syllogism from this and O L,
S L
is obtained.

4. The argument is translated as follows:
G F
Z M
M F
--------------------------------------------
Z G

The inference rules used are:
>From Z M and
M F
by hypothetical syllogism
Z F is obtained.
Then from G F
by taking contrapositive M F is obtained.
>From this and Z F
by hypothetical syllogism Z G is obtained.

5. The argument is translated as follows:
R ( H L )
H ( L R )
--------------------------------------------
R

The inference rules used are:
From H ( L R ) by distributive law
( H L ) ( H R )
From this by simplification
( H L )
From this by De Morgan ( H L )
With this and R ( H L )
by disjunctive syllogism R is concluded.