1 (a)
A | B = A>->T,(B>->T,F)

A & B = A>->(B>->T,F),F

-A = A>->F,T

1 (b)

(p1<->q1) & (p2<->q2)
p1>->((q1) & (p2<->q2)) , ((-q1) & (p2<->q2))
p1>->(q1>-> p2<->q2, F) , (q1>-> F, p2<->q2)
p1>->(q1>-> (p2>->q2,-q2), F) , (q1>-> F, (p2>->q2,-q2))
p1>->(q1>-> (p2>->(q2>->T,F),(q2>->F,T)), F) , (q1>-> F, (p2>->(q2>->T,F),(q2>->F,T)))

1 (c)

F = {{p,q,-r,t},{p,-q,t},{-t},{r},{s}}
{{p,q,-r,t},{p,-q,t},{-t},{r},{s},{-p}}

p>->{{}} , {{q,-r,t},{-q,t},{-t},{r},{s}}
p>->{{}} , t>-> {{}} , {{q,-r},{-q},{r},{s}}
p>->{{}} , t>-> {{}} , q>-> {{}}, {{-r},{r},{s}}
p>->{{}} , t>-> {{}} , q>-> {{}}, r>->{{}} , {{}}

2 (a)

1 {p,q,-r,t} input
2 {p,-q,t} input
3 {-t} input
4 {r} input
5 {s} input
6 {-p} negation of thesis
7 {q,-r,t} 1+6
8 {q,t} 7+4
9 {q} 8+3
10 {-q,t} 2+6
11 {-q} 10+3
12 {} 11+9

2(b)
For example it is not true that
{{p}} |-R {p,q}
although 
{{p}} |= {p,q}
(but it can be proved by refutation)

3(a)
{{p,q,-r,t},{p,-q,t},{-t},{r},{s},{-p}} (unit clause)
{{q,-r,t},{p,-q,t},{-t},{r},{s},{-p}} (unit clause)
{{q,-r,t},{-q,t},{-t},{r},{s}} (unit clause)
{{q,t},{-q,t},{-t},{r},{s}} (unit clause)
{{q},{-q,t},{-t},{r},{s}} (unit clause)
{{q},{-q},{-t},{r},{s}} (unit clause)
{{},{-q},{-t},{r},{s}} (termination rule)

3(b)
F = (G1|L) & ... & (Gm|L) & (H1|-L) & ... & (Hn|-L) & G
Consider an interpretation I such that I|=L, then
 if I|=F then I|=H1 & ... & Hn & G

Consider now an interpretation I such that I|#L, then
 if I|=F then I|=G1 & ... & Gm & G

Consider the set of all interpretations: in any interpretation I, either I|=L or I|#L. Therefore,
 if I|=F then I|=G1 & ... & Gm & G | H1 & ... & Hn & G
q.e.d.