### Premises - Type 2: Some A's are B's

There are three possible Venn Diagrams for this case:

Possible Venn Diagram 1:

In this case, A and B have some elements in common and there are some elements present in each set which do not belong to the other. The colored region represents the elements of A which belong to B as well.

We can conclude the following from this case:

(a) Some B’s are A’s

(b) Some A’s are not B’s: Of course, when we say Some A’s are B’s, there can be some A’s which do not belong to B.

(c) Some B’s are not A’s

Possible Venn Diagram 2:

In this case, all the elements of B are the elements of A as well and in addition to that A consists of some elements which do not belong to B. Therefore, there are only some elements of A which belong to B.

We can conclude the following from this case:

(a) All B’s are A’s

(b) Some A’s are not B’s

Possible Venn Diagram 3:

In this case, all the elements of A belong to B. Therefore, we can say that some elements of A belong to B and hence “Some A’s are B’s”.

We can conclude the following from this case:

(a) All A’s are B’s.

(b) Some B’s are not A’s

(c) Some B’s are A’s

In all we can conclude the following from this premise:

(a) Some B’s are A’s: This is the only definite conclusion which is true for all the cases.

(b) All other conclusions are probable conclusions which are valid under specific conditions as mentioned above. We have listed them below for your reference:

1. Some A’s are not B’s.

2. Some B’s are not A’s.

3. All B’s are A’s.

4. All A’s are B’s.

Let’s practice what we have learned. In the questions below two statements are given and these statements are followed by some conclusions. You have to take the given statements to be true even if they seem to be at variance from commonly known facts. Read the conclusions and then decide which of the given conclusions logically follows from the given statements, disregarding commonly known facts.

Practice Exercise 1:

Statements:

All burgers are noodles.

Some noodles are pizzas.

Conclusions:

I. All noodles are burgers.

II. Some noodles are not pizzas.

III. Some burgers are pizzas.

IV. Some burgers may or may not be pizzas.

Hint: First draw the Venn diagram representation of the statements and then try to identify the correct conclusions.

We can draw the Venn diagram representation of the statements as shown: Now we don’t know the relation between pizzas and burgers, so there is a possibility that some pizzas might be burgers as well. Remember that this is just a possibility. Considering this possibility, we can also draw the Venn diagram as: Now let’s see which of the conclusions are correct:

I. All noodles are burgers.

This is a probable conclusion, that is, there is a possibility that this might be true, but as we can see in the Venn diagrams, it is not a definite conclusion. The correct conclusion would be “Some noodles are burgers”.

II. Some noodles are not pizzas.

As we discussed in the text, this is again a probable conclusion. The correct conclusion would be “Some pizzas are noodles”.

III. Some burgers are pizzas.

Again, this is just a probable conclusion. There is just a possibility that some burgers might be pizzas. The correct conclusion would be  “Some burgers may or may not be pizzas” or “Some pizzas may or may not be burgers”.

IV. Some burgers may or may not be pizzas.

As we have already discussed, this is a definite conclusion.

Therefore, only IV is the correct conclusion.

Note that instead of drawing separate Venn diagrams, we can represent another possibility in the same diagram as the dotted circle: Practice Exercise 2:

Statements:

Some boys are students.

All students are teenagers.

Conclusions:

I. All teenagers are students.

II. Some boys are teenagers.

III. All boys being teenagers is a possibility.

We can draw the Venn diagram representation of the statements as shown: We don’t know the relation between boys and teenagers. So, we have represented another possibility with a dotted circle. Both cases support the fact that “All students are teenagers”.

Now let’s see which of the conclusions are correct:

I. All teenagers are students.

This is a probable conclusion. As we can see in the Venn diagram, it is not a definite conclusion. The correct conclusion would be “Some teenagers are students”.

II. Some boys are teenagers.

As we can see in the Venn diagram, this is a definite conclusion.

III. All boys being teenagers is a possibility.

Yes, there is a possibility that all boys might be teenagers. We can see that in the Venn diagram as well. So, this is also a definite conclusion.

Therefore, both II and III are the correct conclusions.

Practice Exercise 3:

Statements:

Some books are trees.

Some trees are plants.

Conclusions:

I. Some books being plants is a possibility.

II. All trees being plants is a possibility.

We can draw the Venn diagram representation of the statements as shown: We are not given any relation between books and plants. So, there is a possibility that some plants might be books as well. We have represented this possibility with a dotted circle in the Venn diagram.

Now let’s see which of the conclusions are correct:

I. Some books being plants is a possibility.

As we can see in the Venn diagram, some books being plants is a possibility. So, this conclusion follows.

II. All trees being plants is a possibility.

As discussed in the text, since some trees are plants, therefore all trees being plants is a possibility. So, this conclusion follows as well.

Therefore, both I and II are the correct conclusions.

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