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Its a bunch of problems but each of them are “Select the conclusion that follows in a single step from the given premises in each problem. I did the best to format it quickly in the attached document sorry if I could just take a few pictures. They are multiple choice though,

Instructions
INSTRUCTIONS: Select the conclusion that follows in a single step from the given premises in
each problem.
Question 1
Given the following premises:
1. Ã¯ÂÂ¾R Ã¯â€šÂº Ã¯ÂÂ¾R
2. N Ã¢â‚¬Â¢ Ã¯ÂÂ¾T
3. R Ã¯Æ’â€° Ã¯ÂÂ¾(N Ã¢â‚¬Â¢ Ã¯ÂÂ¾T)
R Ã¯Æ’â€° (Ã¯ÂÂ¾N Ã¯Æ’Å¡ Ã¯ÂÂ¾Ã¯ÂÂ¾T) —— ( 3, DM)
Ã¯ÂÂ¾R —— ( 2, 3, MT)
(N Ã¢â‚¬Â¢ Ã¯ÂÂ¾T) Ã¯Æ’â€° Ã¯ÂÂ¾R —— ( 3, Trans)
Ã¯ÂÂ¾R —— (1, Taut)
Ã¯ÂÂ¾T —— (2, Simp)
Question 2
Given the following premises:
1. G Ã¢â‚¬Â¢ Ã¯ÂÂ¾A
2. K Ã¯Æ’â€° (G Ã¢â‚¬Â¢ Ã¯ÂÂ¾A)
3. G Ã¯Æ’â€° M
K —— (1, 2, MP)
K Ã¯Æ’â€° (Ã¯ÂÂ¾A Ã¢â‚¬Â¢ G) —— (2, Com)
M —— (1, 3, MP)
(K Ã¯Æ’â€° G) Ã¢â‚¬Â¢ Ã¯ÂÂ¾A —— (2, Assoc)
(K Ã¯Æ’â€° G ) Ã¯Æ’â€° Ã¯ÂÂ¾A —— (2, Exp)
Question 3
Given the following premises:
1. Ã¯ÂÂ¾(Q Ã¢â‚¬Â¢ Ã¯ÂÂ¾S)
2. Ã¯ÂÂ¾F Ã¯Æ’â€° (Q Ã¢â‚¬Â¢ Ã¯ÂÂ¾S)
3. H Ã¯Æ’Å¡ (Q Ã¢â‚¬Â¢ Ã¯ÂÂ¾S)
H ———- (1, 3, DS)
Ã¯ÂÂ¾Q Ã¯Æ’Å¡ S ——– ( 1, DM)
F ——— (1, 2, MT)
(H Ã¢â‚¬Â¢ Q) Ã¯Æ’Å¡ (H Ã¢â‚¬Â¢ Ã¯ÂÂ¾S) —— ( 3, Dist)
Ã¯ÂÂ¾Ã¯ÂÂ¾F ——– (1, 2, MT)
Question 4
Given the following premises:
1.
N
2.
R Ã¢Å Æ’ ~N
3.
~C Ã‚Â· (T Ã¢Å Æ’ R)
Ã¯ÂÂ¾R ——– (1, 2, MT)
(Ã¯ÂÂ¾C Ã¢â‚¬Â¢ T) Ã¯Æ’â€° R ——— ( 3, Assoc)
N Ã¯Æ’â€° Ã¯ÂÂ¾R ——— ( 2, Trans)
T Ã¯Æ’â€° Ã¯ÂÂ¾N ——– (2, 3, HS)
Ã¯ÂÂ¾C ———- ( 3, Simp)
Question 5
Given the following premises:
1.
(K Ã‚Â· ~T) Ã¯Æ’Å¡ (K Ã‚Â· ~H)
2.
~M Ã¢Å Æ’ (K Ã‚Â· ~H)
3.
~(K Ã‚Â· ~H)
Ã¯ÂÂ¾K Ã¯Æ’Å¡ H ———– (3, DM)
K Ã¢â‚¬Â¢ Ã¯ÂÂ¾T ———– ( 1, 3, DS)
M ——– ( 2, 3, MT)
(Ã¯ÂÂ¾M Ã¢â‚¬Â¢ K) Ã¯Æ’â€° Ã¯ÂÂ¾H —— (2, Exp)
K Ã¢â‚¬Â¢ (Ã¯ÂÂ¾T Ã¯Æ’Å¡ Ã¯ÂÂ¾H) ——- (1, Dist)
Question 6
Given the following premises:
1.
A
2.
G Ã¢Å Æ’ (A Ã¢Å Æ’ ~L)
3.
~A Ã¯Æ’Å¡ ~G
G Ã¯Æ’â€° (Ã¯ÂÂ¾Ã¯ÂÂ¾L Ã¯Æ’â€° Ã¯ÂÂ¾A) ——–(2, Trans)
Ã¯ÂÂ¾G ——— ( 1, 3, DS)
Ã¯ÂÂ¾L ———– ( 1, 2, MP)
(G Ã¯Æ’â€° A) Ã¯Æ’â€° Ã¯ÂÂ¾L ———– (2, Assoc)
A Ã¯Æ’Å¡ G ———– ( 3, DN)
Question 7
Given the following premises:
1.
(S Ã¢Å Æ’ ~F) Ã‚Â· (~F Ã¢Å Æ’ B)
2.
S Ã¯Æ’Å¡ ~F
3.
~F
Ã¯ÂÂ¾F Ã¯Æ’Å¡ B ——— (1, 2, CD)
S ——— ( 2, 3, DS)
Ã¯ÂÂ¾S ———– (1, 3, MT)
B ———— (1, 3, MP)
S Ã¯Æ’â€° B ———- (1, HS)
Question 8
Given the following premises:
1.
N Ã¢â€°Â¡ R
2.
(N Ã‚Â· ~R) Ã¢Å Æ’ C
3.
N
R ——- (1, 3, MP)
N Ã¢â‚¬Â¢ (Ã¯ÂÂ¾R Ã¯Æ’â€° C) ——- ( 2, Assoc)
(N Ã¯Æ’â€° R) Ã¯Æ’Å¡ (R Ã¯Æ’â€° N) ——- (1, Equiv)
C Ã¯Æ’â€° (N Ã¢â‚¬Â¢ Ã¯ÂÂ¾R) ——- ( 2, Com)
N Ã¯Æ’â€° (Ã¯ÂÂ¾R Ã¯Æ’â€° C) ——- (2, Exp)
Question 9
Given the following premises:
1.
~M Ã¢Å Æ’ S
2.
~M
3.
(M Ã¯Æ’Å¡ H) Ã¯Æ’Å¡ ~S
M Ã¯Æ’Å¡ (H Ã¯Æ’Å¡ Ã¯ÂÂ¾S) ——- 3, Assoc
M Ã¯Æ’Å¡ H ——- 3, Simp
Ã¯ÂÂ¾S ——- 1, 2, MP
H ——- 2, 3, DS
M Ã¯Æ’Å¡ S ——- 1, Impl
Question 10
Given the following premises:
1.
(J Ã‚Â· ~N) Ã¯Æ’Å¡ T
2.
~(J Ã‚Â· ~N)
3.
~T
T ——- 1, 2, DS
Ã¯ÂÂ¾J ——- 2, Simp
Ã¯ÂÂ¾J Ã¯Æ’Å¡ N ——- 2, DM
J Ã¢â‚¬Â¢ (Ã¯ÂÂ¾N Ã¯Æ’Å¡ T) ——- 1, Assoc
J Ã¢â‚¬Â¢ Ã¯ÂÂ¾N ——- 1, 3, DS
Question 11
Given the following premises:
1.
~U Ã¢Å Æ’ (S Ã‚Â· K)
2.
R Ã¢Å Æ’ (~U Ã‚Â· ~U)
3.
S Ã¢â€°Â¡ ~U
(S Ã¯Æ’â€° U) Ã¢â‚¬Â¢ (Ã¯ÂÂ¾U Ã¯Æ’â€° Ã¯ÂÂ¾S) ——- 3, Equiv
R Ã¯Æ’â€° U ——- 2, DN
R Ã¯Æ’â€° (S Ã¢â‚¬Â¢ K) ——- 1, 2, HS
R Ã¯Æ’â€° Ã¯ÂÂ¾U ——- 2, Taut
(Ã¯ÂÂ¾U Ã¢â‚¬Â¢ S) Ã¯Æ’â€° K ——- 1, Exp
Question 12
Given the following premises:
1.
~I Ã¯Æ’Å¡ ~~B
2.
M Ã¢Å Æ’ ~I
3.
I
Ã¯ÂÂ¾I Ã¯Æ’â€° M ——- 2, Com
Ã¯ÂÂ¾M ——- 2, 3, MT
Ã¯ÂÂ¾Ã¯ÂÂ¾B ——- 1, 3, DS
Ã¯ÂÂ¾(I Ã¢â‚¬Â¢ Ã¯ÂÂ¾B) ——- 1, DM
M Ã¯Æ’â€° Ã¯ÂÂ¾Ã¯ÂÂ¾B ——- 1, 2, HS
Question 13
Given the following premises:
1.
~N Ã‚Â· ~F
2.
K Ã¢Å Æ’ (N Ã‚Â· F)
3.
U Ã¯Æ’Å¡ (K Ã‚Â· ~N)
(U Ã¯Æ’Å¡ K) Ã¢â‚¬Â¢ Ã¯ÂÂ¾N ——- 3, Assoc
Ã¯ÂÂ¾(N Ã¢â‚¬Â¢ F) ——- 1, DM
Ã¯ÂÂ¾K ——- 1, 2, MT
(U Ã¯Æ’Å¡ K) Ã¢â‚¬Â¢ (U Ã¯Æ’Å¡ Ã¯ÂÂ¾N) ——- 3, Dist
(K Ã¢â‚¬Â¢ N) Ã¯Æ’â€° F ——- 2, Exp
Question 14
Given the following premises:
1.
DÃ¢Å Æ’ H
2.
~D
3.
~(D Ã¯Æ’Å¡ S)
Ã¯ÂÂ¾D Ã¯Æ’Å¡ Ã¯ÂÂ¾S ——- 3, DM
Ã¯ÂÂ¾H ——- 1, 2, MT
S ——- 2, 3, DS
Ã¯ÂÂ¾D Ã¯Æ’Å¡ (D Ã¯Æ’â€° H) ——- 2, Add
H Ã¯Æ’â€° D ——- 1, Com
Question 15
Given the following premises:
1.
A
2.
(A Ã¢Å Æ’ ~T) Ã¢Å Æ’ ~G
3.
Q Ã¢Å Æ’ (A Ã¢Å Æ’ ~T)
Q Ã¯Æ’â€° (T Ã¯Æ’â€° Ã¯ÂÂ¾A) ——- 3, Trans
Ã¯ÂÂ¾T ——- 1, 3, MP
(Q Ã¯Æ’â€° A) Ã¯Æ’â€° Ã¯ÂÂ¾T ——- 3, Assoc
A Ã¯Æ’â€° (Ã¯ÂÂ¾T Ã¢â‚¬Â¢ Ã¯ÂÂ¾G) ——- 2, Exp
Q Ã¯Æ’â€° Ã¯ÂÂ¾G ——- 2, 3, HS
Question 16
Given the following premises:
1.
P Ã‚Â· (~H Ã¯Æ’Å¡ D)
2.
~(~P Ã‚Â· ~H)
3.
(P Ã¢Å Æ’ ~H) Ã‚Â· (~P Ã¢Å Æ’ H)
(P Ã¢â‚¬Â¢ Ã¯ÂÂ¾H) Ã¯Æ’Å¡ D ——- 1, Assoc
P Ã¢â‚¬Â¢ (H Ã¯Æ’â€° D) ——- 1, Impl
Ã¯ÂÂ¾H Ã¯Æ’Å¡ D ——- 1, Simp
P Ã¯â€šÂº Ã¯ÂÂ¾H ——- 3, Equiv
P Ã¢â‚¬Â¢ H ——- 2, DN
Question 17
Given the following premises:
1.
NÃ¯Æ’Å¡ C
2.
(N Ã¯Æ’Å¡ C) Ã¢Å Æ’ (F Ã¢Å Æ’ C)
3.
~C
Ã¯ÂÂ¾N ——- 1, 3, MT
F Ã¯Æ’â€° C ——- 1, 2, MP
Ã¯ÂÂ¾F ——- 2, 3, MT
N ——- 1, 3, DS
Ã¯ÂÂ¾C Ã¢â‚¬Â¢ R ——- 3, Add
Question 18
Given the following premises:
1.
(S Ã‚Â· ~J) Ã¯Æ’Å¡ (~S Ã‚Â· ~~J)
2.
S Ã¯Æ’Å¡ ~S
3.
~J Ã¢Å Æ’ P
Ã¯ÂÂ¾P Ã¯Æ’â€° J ——- 3, Trans
J Ã¯Æ’Å¡ P ——- 3, Impl
S Ã¯â€šÂº Ã¯ÂÂ¾J ——- 1, Equiv
Ã¯ÂÂ¾J Ã¯Æ’Å¡ Ã¯ÂÂ¾Ã¯ÂÂ¾J ——- 1, 2, CD
S ——- 2, Taut
Question 19
Given the following premises:
1.
Q Ã¢Å Æ’ (A Ã¯Æ’Å¡ ~T)
2.
T
3.
A Ã¯Æ’Å¡ ~T
Q ——- 1, 3, MP
Q Ã¯Æ’â€° (Ã¯ÂÂ¾Ã¯ÂÂ¾A Ã¯Æ’Å¡ Ã¯ÂÂ¾T) ——- 1, DN
A ——- 2, 3, DS
(Q Ã¯Æ’â€° A) Ã¯Æ’Å¡ Ã¯ÂÂ¾T ——- 1, Assoc
(A Ã¯Æ’Å¡ Ã¯ÂÂ¾T) Ã¯Æ’â€° Q ——- 1, Com
Question 20
Given the following premises:
1.
~E Ã¢Å Æ’ P
2.
~P
3.
~(P Ã¯Æ’Å¡ ~H)
Ã¯ÂÂ¾H ——- 2, 3, DS
Ã¯ÂÂ¾P Ã¢â‚¬Â¢ Ã¯ÂÂ¾(P Ã¯Æ’Å¡ Ã¯ÂÂ¾H) ——- 2, 3, Conj
Ã¯ÂÂ¾P Ã¯Æ’â€° E ——- 1, Trans
E ——- 1, 2, MT
Ã¯ÂÂ¾P Ã¢â‚¬Â¢ H ——- 3, DM
Question 21
Given the following premises:
1. Ã¯ÂÂ¾Ã¯ÂÂ¾N
2. K Ã¯Æ’â€° Ã¯ÂÂ¾N
3. Ã¯ÂÂ¾N Ã¯Æ’Å¡ (K Ã¢â‚¬Â¢ S)
(Ã¯ÂÂ¾N Ã¯Æ’Å¡ K) Ã¢â‚¬Â¢ S 3, Assoc
K 1, 2, MT
(Ã¯ÂÂ¾N Ã¢â‚¬Â¢ K) Ã¯Æ’Å¡ (Ã¯ÂÂ¾N Ã¢â‚¬Â¢ S) 3, Dist
K Ã¢â‚¬Â¢ S 1, 3, DS
N Ã¯Æ’â€° Ã¯ÂÂ¾K 2, Trans
Question 22
Given the following premises:
1. Q Ã¯Æ’â€° (H Ã¢â‚¬Â¢ Ã¯ÂÂ¾F)
2. Ã¯ÂÂ¾(Q Ã¢â‚¬Â¢ Ã¯ÂÂ¾M)
3. Ã¯ÂÂ¾G Ã¯Æ’â€° (Q Ã¢â‚¬Â¢ Ã¯ÂÂ¾M)
Q Ã¯Æ’â€° Ã¯ÂÂ¾(Ã¯ÂÂ¾H Ã¯Æ’Å¡ F) 1, DM
G Ã¯Æ’Å¡ Ã¯ÂÂ¾(Q Ã¢â‚¬Â¢ M) 2, Add
G 2, 3, MT
Ã¯ÂÂ¾Q Ã¯Æ’Å¡ Ã¯ÂÂ¾Ã¯ÂÂ¾M 2, DM
Q 2, Simp
Question 23
Given the following premises:
1. (S Ã¯Æ’â€° R) Ã¯Æ’â€° (J Ã¯Æ’â€° T)
2. (P Ã¯Æ’â€° R) Ã¯Æ’â€° (S Ã¯Æ’â€° R)
3. R Ã¯Æ’â€° J
P Ã¯Æ’â€° J 2, 3, HS
R Ã¯Æ’â€° T 1, 3, HS
(P Ã¯Æ’â€° R) Ã¯Æ’â€° (J Ã¯Æ’â€° T) 1, 2, HS
S Ã¯Æ’â€° J 1, 3, HS
(S Ã¯Æ’â€° R) Ã¢â‚¬Â¢ (P Ã¯Æ’â€° R) 1, 2, Conj
Question 24
Given the following premises:
1. Q Ã¯Æ’â€° (H Ã¢â‚¬Â¢ L)
2. H Ã¯Æ’â€° Ã¯ÂÂ¾Q
3. L Ã¯Æ’â€° Ã¯ÂÂ¾Q
Q Ã¯Æ’â€° Ã¯ÂÂ¾Q 1, 3, HS
H Ã¯Æ’â€° L 2, 3, HS
(Q Ã¯Æ’â€° H) Ã¯Æ’â€° L 1, Exp
(L Ã¯Æ’â€° Ã¯ÂÂ¾Q) Ã¢â‚¬Â¢ (H Ã¯Æ’â€° Ã¯ÂÂ¾Q) 2, 3, Conj
L Ã¯Æ’â€° (H Ã¢â‚¬Â¢ L) 1, 3, HS
Question 25
Given the following premises:
1. F Ã¯Æ’â€° J
2. A Ã¯Æ’â€° (F Ã¢â‚¬Â¢ J)
3. A Ã¢â‚¬Â¢ (Q Ã¯Æ’Å¡ N)
(A Ã¯Æ’â€° F) Ã¢â‚¬Â¢ (A Ã¯Æ’â€° J) 2, Dist
A Ã¢â‚¬Â¢ (N Ã¯Æ’Å¡ Q) 3, Com
(A Ã¢â‚¬Â¢ Q) Ã¯Æ’Å¡ N 3, Assoc
A Ã¯Æ’â€° J 1, 2, HS
J Ã¯Æ’â€° F 1, Com
Question 26
Given the following premises:
1. Ã¯ÂÂ¾D Ã¯Æ’Å¡ Ã¯ÂÂ¾T
2. D Ã¯Æ’Å¡ (Ã¯ÂÂ¾T Ã¢â‚¬Â¢ Ã¯ÂÂ¾R)
3. D
(D Ã¯Æ’Å¡ Ã¯ÂÂ¾T) Ã¢â‚¬Â¢ (D Ã¯Æ’Å¡ Ã¯ÂÂ¾R) 2, Dist
Ã¯ÂÂ¾T Ã¢â‚¬Â¢ Ã¯ÂÂ¾R 2, 3, DS
(D Ã¯Æ’Å¡ Ã¯ÂÂ¾T) Ã¢â‚¬Â¢ R 2, Assoc
Ã¯ÂÂ¾T 1, 3, DS
D Ã¯Æ’Å¡ T 1, DN
Question 27
Given the following premises:
1. (C Ã¢â‚¬Â¢ Ã¯ÂÂ¾F) Ã¯Æ’â€° E
2. G Ã¯Æ’Å¡ (C Ã¢â‚¬Â¢ Ã¯ÂÂ¾F)
3. Ã¯ÂÂ¾(C Ã¢â‚¬Â¢ Ã¯ÂÂ¾F)
G Ã¯Æ’â€° E 1, 2, HS
G 2, 3, DS
C Ã¯Æ’â€° (Ã¯ÂÂ¾F Ã¯Æ’â€° E) 1, Exp
(G Ã¯Æ’Å¡ C) Ã¢â‚¬Â¢ Ã¯ÂÂ¾F 2, Assoc
C 1, Simp
Question 28
Given the following premises:
1. K Ã¯Æ’Å¡ Ã¯ÂÂ¾H
2. (K Ã¯Æ’Å¡ Ã¯ÂÂ¾H) Ã¯Æ’â€° (B Ã¯Æ’â€° J)
3. J Ã¯Æ’â€° D
K 1, Simp
D Ã¯Æ’â€° J 3, Trans
B Ã¯Æ’â€° D 2, 3, HS
B Ã¯Æ’â€° J 1, 2, MP
H Ã¯Æ’â€° K 1, Impl
Question 29
Given the following premises:
1. R Ã¢â‚¬Â¢ Ã¯ÂÂ¾S
2. R Ã¯Æ’â€° Ã¯ÂÂ¾(S Ã¢â‚¬Â¢ Ã¯ÂÂ¾F)
3. Ã¯ÂÂ¾S Ã¯Æ’â€° (F Ã¢â‚¬Â¢ N)
F Ã¢â‚¬Â¢ N 1, 3, MP
R Ã¯Æ’â€° (Ã¯ÂÂ¾S Ã¯Æ’Å¡ Ã¯ÂÂ¾Ã¯ÂÂ¾F) 2, DM
(Ã¯ÂÂ¾S Ã¯Æ’â€° F) Ã¢â‚¬Â¢ (Ã¯ÂÂ¾S Ã¯Æ’â€° N) 3, Dist
Ã¯ÂÂ¾S 1, Simp
(Ã¯ÂÂ¾S Ã¢â‚¬Â¢ F) Ã¯Æ’â€° N 3, Exp
Question 30
Given the following premises:
1. Ã¯ÂÂ¾T Ã¯Æ’â€° E
2. Ã¯ÂÂ¾K Ã¯Æ’â€° (Ã¯ÂÂ¾T Ã¯Æ’Å¡ Ã¯ÂÂ¾T)
3. M Ã¯Æ’â€° (Ã¯ÂÂ¾K Ã¯Æ’Å¡ Ã¯ÂÂ¾L)
Ã¯ÂÂ¾K Ã¯Æ’â€° E 1, 2, HS
M Ã¯Æ’â€° (K Ã¯Æ’â€° Ã¯ÂÂ¾L) 3, Impl
(M Ã¯Æ’â€° Ã¯ÂÂ¾K) Ã¯Æ’Å¡ L 3, Assoc
Ã¯ÂÂ¾K Ã¯Æ’â€° T 2, Taut
M Ã¯Æ’â€° (K Ã¯Æ’Å¡ L) 3, DN
Question 31
Given the following premises:
1. Ã¯ÂÂ¾R Ã¯Æ’Å¡ Ã¯ÂÂ¾R
2. R Ã¯Æ’Å¡ (Ã¯ÂÂ¾J Ã¢â‚¬Â¢ Ã¯ÂÂ¾H)
3. Ã¯ÂÂ¾R Ã¯Æ’â€° (H Ã¢â‚¬Â¢ B)
R 1, DN
(R Ã¯Æ’Å¡ Ã¯ÂÂ¾J) Ã¢â‚¬Â¢ Ã¯ÂÂ¾H 2, Assoc
H Ã¢â‚¬Â¢ B 1, 3, MP
Ã¯ÂÂ¾J Ã¢â‚¬Â¢ Ã¯ÂÂ¾H 1, 2, DS
R Ã¯Æ’Å¡ Ã¯ÂÂ¾(J Ã¯Æ’Å¡ H) 2, DM
Question 32
Given the following premises:
1. (F Ã¢â‚¬Â¢ Ã¯ÂÂ¾M) Ã¯Æ’â€° (L Ã¢â‚¬Â¢ Ã¯ÂÂ¾G)
2. P Ã¯Æ’â€° L
3. Ã¯ÂÂ¾(L Ã¢â‚¬Â¢ Ã¯ÂÂ¾G)
Ã¯ÂÂ¾L Ã¯Æ’Å¡ G 3, DM
Ã¯ÂÂ¾(F Ã¢â‚¬Â¢ Ã¯ÂÂ¾M) 1, 3, MT
L Ã¯Æ’â€° P 2, Trans
Ã¯ÂÂ¾P 2, 3, MT
Ã¯ÂÂ¾L 3, Simp
Question 33
Given the following premises:
1. S Ã¯Æ’â€° (Ã¯ÂÂ¾Ã¯ÂÂ¾T Ã¢â‚¬Â¢ Ã¯ÂÂ¾Ã¯ÂÂ¾C)
2. (S Ã¢â‚¬Â¢ Q) Ã¯Æ’Å¡ C
3. Ã¯ÂÂ¾C
S Ã¢â‚¬Â¢ Q 2, 3, DS
S Ã¯Æ’â€° (T Ã¢â‚¬Â¢ Ã¯ÂÂ¾Ã¯ÂÂ¾C) 1, DN
S 2, Simp
S Ã¯Æ’â€° Ã¯ÂÂ¾Ã¯ÂÂ¾T 1, Simp
S Ã¯Æ’â€° (T Ã¢â‚¬Â¢ C) 1, DN
Question 34
Given the following premises:
1. (Ã¯ÂÂ¾H Ã¢â‚¬Â¢ Ã¯ÂÂ¾J) Ã¯Æ’â€° K
2. Ã¯ÂÂ¾(Ã¯ÂÂ¾H Ã¢â‚¬Â¢ Ã¯ÂÂ¾J)
3. (Ã¯ÂÂ¾H Ã¢â‚¬Â¢ N) Ã¯Æ’Å¡ (Ã¯ÂÂ¾H Ã¢â‚¬Â¢ Ã¯ÂÂ¾J)
Ã¯ÂÂ¾H Ã¯Æ’â€° (J Ã¯Æ’â€° K) 1, Exp
Ã¯ÂÂ¾H Ã¢â‚¬Â¢ (N Ã¯Æ’Å¡ Ã¯ÂÂ¾J) 3, Dist
(Ã¯ÂÂ¾H Ã¢â‚¬Â¢ N) Ã¯Æ’â€° K 1, 3, HS
Ã¯ÂÂ¾H Ã¢â‚¬Â¢ N 2, 3, DS
H Ã¯Æ’Å¡ J 2, DM
Question 35
Given the following premises:
1. R Ã¯Æ’â€° (Ã¯ÂÂ¾B Ã¯Æ’â€° F)
2. Ã¯ÂÂ¾U Ã¯Æ’â€° B
3. Ã¯ÂÂ¾B
F 1, 3, MP
R Ã¯Æ’â€° (Ã¯ÂÂ¾F Ã¯Æ’â€° Ã¯ÂÂ¾Ã¯ÂÂ¾B) 1, Trans
(R Ã¯Æ’â€° Ã¯ÂÂ¾B) Ã¯Æ’â€° F 1, Assoc
U 2, 3, MT
Ã¯ÂÂ¾B Ã¯Æ’â€° U 2, Trans
Question 36
Given the following premises:
1. T Ã¯Æ’Å¡ S
2. A Ã¯Æ’â€° T
3. A Ã¢â‚¬Â¢ (Ã¯ÂÂ¾T Ã¢â‚¬Â¢ S)
S 1, 3, DS
T Ã¯Æ’â€° A 2, Com
(A Ã¢â‚¬Â¢ Ã¯ÂÂ¾T) Ã¢â‚¬Â¢ S 3, Assoc
Ã¯ÂÂ¾T 3, Simp
T 2, 3, MP
Question 37
Given the following premises:
1. Ã¯ÂÂ¾P
2. L Ã¯Æ’â€° (P Ã¯Æ’Å¡ M)
3. (P Ã¢â‚¬Â¢ M) Ã¯Æ’â€° (Ã¯ÂÂ¾R Ã¯Æ’Å¡ Ã¯ÂÂ¾R)
L Ã¯Æ’â€° (Ã¯ÂÂ¾R Ã¯Æ’Å¡ Ã¯ÂÂ¾R) 2, 3, HS
P 3, Simp
M 1, 2, DS
(P Ã¢â‚¬Â¢ M) Ã¯Æ’â€° Ã¯ÂÂ¾R 3, Taut
(L Ã¯Æ’â€° P) Ã¯Æ’Å¡ (L Ã¯Æ’â€° M) 2, Dist
Question 38
Given the following premises:
1. Ã¯ÂÂ¾N Ã¯Æ’Å¡ H
2. Q Ã¯Æ’â€° Ã¯ÂÂ¾(Ã¯ÂÂ¾N Ã¯Æ’Å¡ H)
3. (Ã¯ÂÂ¾N Ã¯Æ’â€° Q) Ã¢â‚¬Â¢ (H Ã¯Æ’â€° Q)
Q Ã¯Æ’Å¡ Q 1, 3, CD
Ã¯ÂÂ¾Q 1, 2, MT
Q Ã¯Æ’â€° (N Ã¢â‚¬Â¢ Ã¯ÂÂ¾H) 2, DM
Ã¯ÂÂ¾N Ã¯Æ’â€° Ã¯ÂÂ¾(Ã¯ÂÂ¾N Ã¯Æ’Å¡ H) 2, 3, HS
H Ã¯Æ’â€° Q 3, Simp
Question 39
Given the following premises:
1. R Ã¯Æ’â€° (E Ã¢â‚¬Â¢ D)
2. R Ã¢â‚¬Â¢ Ã¯ÂÂ¾G
3. Ã¯ÂÂ¾E Ã¯Æ’â€° G
E Ã¯Æ’Å¡ G 3, Impl
Ã¯ÂÂ¾G 2, Simp
(R Ã¢â‚¬Â¢ Ã¯ÂÂ¾G) Ã¯Æ’Å¡ F 2, Add
Ã¯ÂÂ¾Ã¯ÂÂ¾E 2, 3, MT
E Ã¢â‚¬Â¢ D 1, 2, MP
Question 40
Given the following premises:
1. (L Ã¯Æ’â€° M) Ã¢â‚¬Â¢ (F Ã¯Æ’â€° J)
2. M Ã¯Æ’â€° Ã¯ÂÂ¾(F Ã¯Æ’Å¡ L)
3. F Ã¯Æ’Å¡ L