Propositional calculus - I can't get why the answer for this test question is what it is Consider the following premises.


*

*If A = B then B = C.

*B != C.

*If C > D then D < E.

*F != G and A = B.

*A = B or C > D.


Alternatives:
a) F != G
b) F != G and D < E
c) A = B
d) B = C or D < E
e) D < E
The answer sheet says the correct alternative is B.
I am confused, because (1) and (2) seems to imply that A != B. In which case (4) would be contradictory. That's why I am wondering if the question itself is wrong.
 A: I suspect that there is a typo on your source's part, namely being that (4) should be a disjunction. 
Making this correction, we have
(6) $A\neq B$. Take the contrapositive of (1) and applying modus ponens with (2).
(7) $C>D$. Use disjunctive syllogism on (5) with (6).
(8) $D<E$. Apply modus ponens on (3) with (7).
(9) $F\neq G$. Apply disjunctive syllogism on (modified 4) with (6).
(10) b) Conjunction between (8) and (9).
Thus b) would then indeed be a correct answer.
A: You are right; the question is wrong. From (4) we get A = B, and then by (1) we get B = C, contradicting (2).
A: (1) and (2) do not imply that A != B. Indeed is an if, so that IF  A = B, then we have that A = B = C. However in 2, it is not stated that A=B, we only know that B != C,which implies that A != B.
To prove the equality we see that:
 from 4) F!=G and A=B, which implies 5) A=B or C > D.
Hence from 3) we get the answer b!
Notice that the passage from 5 to 3 works as 'or' works when one of the propositions or both are true, so we can take  C > D to be true.
