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Can someone please help me with the approach to solve similar reasoning questions
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I'm not sure this is a real LSAT question, since there are some flaws in the writing. However, it is testing a common structure that is tested much more directly on the LSAT than on the GMAT. All the same, here's the idea:

Let's say you have a conditional premise of the form "If A, then B," or A-->B. For instance "If it's raining, there must be clouds." What can you conclude from this? Can you go backwards against the arrow and get B-->A? Is it true that if there are clouds, it's raining? No, this is what is called an illegal reversal. That's the error committed in the original argument, and (sort of) in answer choice A, although the answer never actually says that if there were a cat, I would sneeze.

How about a negation? If it's NOT raining, can we conclude that there are NO clouds? No, so we can't get NOT A --> NOT B. This is called an illegal negation.

So what CAN we validly conclude from a conditional statement? If we flip AND negate, we get NOT B--> NOT A, and this must actually be true. If there are NO clouds, it must NOT be raining. This is called the contrapositive.

Short version: Any time you see a conditional statement (A-->B), you can logically conclude the contrapositive (NOT B --> NOT A). In fact, the two statements mean the same thing.
You CANNOT logically conclude the reversal (B-->A) or the negation (NOT A-->NOT B). Those two also mean the same thing as each other, and often show up in faulty logic and trap answers.

(Also, for the record, notice that answers B,C, and E are all VALID applications of a conditional or its contrapositive. D just doesn't give us enough information to work, but it doesn't have the same reversal flaw as the original.)
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­If the rebels truly want a political settlement, they will stop shelling the Capitol. They did stop shelling the Capitol. Hence, the rebels sincerely want peace.

Let's understand it
If political settlement --> stop shelling

means 
1. political settlement --implies -- stopped shelling
2. Did not stop shelling --implies -- no political settlement. 

doesn't mean (wrong)
1. no political statement --implies -- no stop shelling 
2. stop shelling -- implies --political settlement - the argument "They did stop shelling the Capitol. Hence, the rebels sincerely want peace" uses this "flawed" logic. We need to find an option with such flawed logic. 

Which one of the following uses reasoning that is most similar to that used in the above argument?

Option Elimination - 

(A) There’s a cat in this house. There must be, because I’m allergic to cats; and I just sneezed.

Meaning - 
If there is a cat in the house, then I can have an allergic reaction. I just had an allergic reaction (sneezed); there is a cat in the house. 

Unpack "If there is a cat in the house, then I can have an allergic reaction."
What it means
1. Cat in the house --implies--an allergic reaction
2. no allergic reaction --implies --no cat

What it doesn't mean (flawed)
1. Allergic reaction --implies -- cat in the house (This option follows this flawed logic "I just had an allergic reaction (sneezed); there is a cat in the house. "
2. No cat -- implies -- no allergic reaction. 

Ok. This is the flawed logic we need same as the flawed logic in the argument. 

(B) In order for a bill to pass, it must be supported by the President. The Crime Bill is not being supported by the President. Therefore, it will not pass.

Necessary condition - Only if supported by the president the bill will pass.
Sufficient condition - If the bill passes, then it must be supported by the president.

What it implies
1. bill passes --implies --supported by the president
2. not supported by the president --implies --the bill will not pass. - This option follows this correct logic. We need flawed logic. Wrong. 

(C) The flood of refugees will continue unless the U.N. sends in peace keepers. The U.N. has announced that peace-keeping troops will not be sent, so the flood of refugees will continue.

Necessary condition  - sending peacekeepers is the minimum condition for no flood of refugees. 
Sufficient condition - If there is no flood of refugees, then the UN must have sent peacekeepers. 

What it implies 
1. no flood of refugees --implies -- UN must have sent peacekeepers
2. No peace keepers -- implies -- flood of refugees. This option follows this correct logic. We need flawed logic. Wrong. 

(D) Every time the United States attempts to mediate between two warring parties, it becomes the target of both. So the same will occur with the civil war in Girunda.

If US mediates, it becomes target. 

Same will occur? same what? becomes target or mediates? Out of scope. 

(E) If you want dessert, you must eat your vegetables. You did not eat your vegetables, so there will be no dessert for you.

If want dessert then eat vegetables 

means no eat vegetables -- no dessert. This option follows this correct logic. We need flawed logic. Wrong. 
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