# Solving multiple statement problems with help of Euler Diagram - Understanding with Example

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Here is one of the examples that we will be discussing in detail. This problem is a multiple statement problem and cannot be solved by the way we have solved the various syllogism problems. These problems can be solved by means of Euler diagram. We will be creating all possible options to solve this problem.

Statements:

Some bats are balls.

No Stars are moons.

All balls are moons.

Conclusions:

1. Some moons are not balls.
2. No balls are stars.
3. Some moons are not bats.
4. Some balls are bats.

We have to find which conclusion follows?

Lets Start Solving this systematically. The idea is instead of making one single Euler Diagram we will be drawing all possible Euler diagrams which can represent the given premises. That is we cannot conclusively say that any one possible Euler Diagram is exactly representing all the premises however at the same time any one of the following Euler diagram is possible with the given premises.

In the first case we take just one premise “All balls are moons” and draw the two possible Euler diagrams. Note that the premise is true for either of the Euler Diagram. In Case 2, all balls are moons and at same time all moons are also balls (that is set of balls is same as set of moons). This case 2 represents the possibility not precluded by the premise.

In this case we add the second premise. So the premises we consider are “No Stars are moons” and “All balls are moons”.

When we consider all the three premises together that is “Some bats are balls”, ” No Stars are moons” and “All balls are moons” we can arrive at all the below Euler Diagrams. To make it clear, case 1 - A represents that bats intersects only the balls that is a bat cannot be moon without also being a ball. Similarly you can understand all the remaining cases.

Now we add bats to the Case 2 and come up with following diagrams. Here Case 2 - A represents a scenario in which bats, moons and balls overlap.

Now lets consider the conclusions one by one. Note that for a conclusion to be absolutely true it must be true with all the above Euler Diagrams. That is to make the conclusion false, we need to find only a single Euler Diagram (derived from the premises) which invalidates it.

Conclusions:

1. Some moons are not balls. - Any Euler Diagram derived from case - 2 invalidates this conclusion. In all these diagrams there are no moons outside the region of balls.
2. No balls are stars. - This conclusion is true as it is valid in all the above Euler Diagrams. Based on the premises, there is not a single possibility where balls can be stars.
3. Some moons are not bats. - Case 2 - A invalidates this conclusion. In this case, all the moons are also bats. That is there are no bats that are also not moon.
4. Some balls are bats. - This conclusion is true across all the possibilities derived from the given premises.

Therefore conclusion 2 and 4 are correct.

Now in general to solve these questions really quickly, we would start start from the conclusions and for each conclusion we will try to come up with a single possibility (from the premises) which invalidates that conclusion. A conclusion is true if there is no possibility which can invalidate it.