Premises - Type 4: Some A's are not B's

There are three possible Venn Diagrams for this case:  

Venn Diagram representing 1st case
Venn Diagram representing 1st case

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 coloured region represents the elements of A which do not belong to B.

 We can conclude the following from this case: 

(a) Some A’s are B’s

(b) Some B’s are not A’s

Venn Diagram representing 2nd case

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.

We can conclude the following from this case: 

(a) Some A’s are B’s

(b) All B’s are A’s

Venn Diagram representing 3rd case

In this case, no element of A belongs to B. Therefore, we can say that some elements of A do not belong to B and hence “Some A’s are not B’s”.

We can conclude the following from this case: 

(a) No A is B.

(b) No B is A.

In all we see that we do not have any definite conclusion from this premise. 

Your Turn:

Let’s practice what we have learnt. In the questions below some 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: 

Some Cricket are Football.

Some Cricket are not Tennis. 

Conclusions:

I. Some Football are Tennis.

II. No Tennis is Football.

We can draw the Venn diagram representation of the statements as shown:  

Venn Diagram Representation of Practice Exercise 1

To solve these problems, it is good to consider all the three cases associated with the premise “Some A’s are not B’s”.

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

I. Some Football are Tennis.

This is a probable conclusion. As we can see in the Venn diagram, it excludes the possibility of no Tennis being a Football. The valid conclusion would be “Some Football being Tennis is a possibility”.

II. No Tennis is Football.

Again, this is a probable conclusion. It excludes the possibility of some Football might be Tennis. So, this conclusion does not follow as well.

Clearly, both the conclusions are probable as well as complement each other. So, either of these conclusions follow.

Practice Exercise 2:

Statements: 

All apples are oranges.

Some oranges are not bananas. 

Conclusions:

I. Some apples being bananas is a possibility.

II. All bananas being apples is a possibility.

We can draw the Venn diagram representation of the statements as shown:  

Venn Diagram Representation of Practice Exercise 2

To solve these problems, it is good to consider all the three cases associated with the premise “Some A’s are not B’s”. Since we don’t know the relation between apples and bananas, we need to consider other possibilities as well. We have represented these possibilities using dotted circles in the Venn Diagram.

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

I. Some apples being bananas is a possibility.

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

II. All bananas being apples is a possibility.

Again, we can see in the Venn Diagram, all bananas being apples is also a possibility. So, this conclusion follows as well.

Therefore, both the conclusions follow.

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