MODULE: Venn diagrams

TUTORIAL V03: Existence

V03.1 Ticks and existence

We have seen how to use shading to indicate that there is nothing in the class represented by the shaded region. We now see how to use ticks to indicate existence.

The basic idea is that when a tick is present in a region, it indicates that there is something in the class represented by the region.

So for example, in the following diagrams, we have a tick outside the circles. Since the area outside the circle represents the class of things that are neither A, nor B, nor C, both diagrams are saying that something exists that is neither A nor B nor C :

There are two impotant points to remember :

A tick in a region says that there is something in the class represented by the region. It does not say how many things there are in that class. There might be just one, or perhaps there are many.
A region without a tick does not represent an empty class. Without a tick, a blank region provides no information as to whether anything exists in the class it represents. Only when a region is shaded can we say that it represents an empty class.

What about the following diagram? What does it represent?

The diagram above does not say "something is A". Actually it says something more specific, namely that "something is A but is not B and not C". If you have given the wrong answer, you might be thinking that the tick indicates that there is something in the class represented by the A circle. But what actually is the case is that a tick is used to indicate existence in the class represented by the smallest bounded region that encloses the tick. In the top two diagrams of this page the smallest bounded area that encloses the tick is the area outside the three circles. In the diagram above, although the purple circle does enclose the tick, it is not the smallest bounded area that does that. That region is actually the one highlighted in this diagram :

V03.2 Exercises

Now see if you can determine what what these diagrams indicate. Click the images for answers.

V03.3 Crossing boundaries

We now come to something slightly more complicated, which is putting a tick across two or more closed regions, as in the following diagram :

However, the interpretation of this digram employs the same rule as before. What the tick indicates is that there is something in the smallest closed region that encloses the tick.

If you compare the above diagram with the one below, you can see that region 1 does not enclose the tick completely, and neither does region 2. The C circle does enclose the tick, but it is not the smallest closed region that does that. Instead it is the combination of region 1 and 2 (a half-moon shape) that is the smallest closed region enclosing the tick.

So what the tick indicates in this diagram is that there is something in the class represented by that region that is made up from 1 and 2. That area of course represents things that are C but not A. In other words, the diagram is saying that something is C but is not A.

Notice that this information does not tell us whether the thing (or things) that exists is B or not. By putting part of the tick inside the B circle and part of it outside, we are leaving it open whether the thing that exists is a B or not.

What do you think is the difference between the diagram above and the one below?

Here we have two ticks, the left one showing that something is B and C but not A, the other one on the right showing that something that is not B and not A is C. So unlike the earlier diagram, this one actually shows that there are at least two distinct things which are C.

V03.4 Exercises

It's now your turn. See if you know what these diagrams represent. Again, click the images for answers.

V03.5 Further discussion

The ticks in the four diagrams above all cut across only one single boundary. We can of course have ticks that cut across more than one boundary, as in this example :

What do you think this means? Applying the same rule of interpretation as before, we see that the smallest closed region that encloses the big tick would have to be the combined three regions which the tick spreads across. This combined region represents things which are either B or C, but which are not A. So what the diagram says is that there is something of this sort.

Finally, if we want to represent the fact that something is A, this is the diagram to use :

Now we can combine what we have learnt about ticks and shading together. Suppose we start with the information that something is both A and C. We therefore draw the following diagram :

Now suppose we are also told that every C is a B. So we add the additional information by shading the appropriate area, and end up with this diagram :

How should this be interpreted and what should we conclude? Half of the green tick is in a shaded region. What does that mean? Give yourself a minute to think about it before you read on ...

The answer is actually quite simple. The tick indicates that something is both A and C. Notice that the tick occupies two separate regions. The left hand side region represents things that are A, B and C. The right hand side region represents things that are A and C but not B. Since the tick crosses these two regions, it indicates that there is something either in the class represented by the left region or in the class represented by the right region (or both of course). Shading tells us that there is nothing in the class represented by the right region. So whatever that exists according to the tick must be in the class represented by the left region. In other words, we can conclude that something is A, B, and C. In effect then, shading "moves" the tick into the left region since it tells us that there is nothing on the right. The above diagram is therefore equivalent to this one :

So here is a general principle you should remember:

A truncated tick within a region R counts as a complete tick in R if part of the tick is in R and all other parts not in R are in shaded regions.

V03.6 Exercises

	? Question 1 - The statement "everything is A" is consistent with this diagram. True or false? [Show answer] True. It is possible that no objects exist in those regions inside the A circle which do not contain a tick.
	? Question 2 - Is the statement "something is either B or C" true or false according to the diagram? Or is there not enough information to decide? [Show answer] The statement is true according to the diagram.
	? Question 3 - "Everything is B or not C" is consistent with the diagram. True or false? [Show answer] False. Do you know Why?
	? Question 4 - Is the statement "nothing is C" true according to the diagram? [Show answer] Not enough information to decide. The C circle has not been shaded, and there are no complete ticks inside the circle.
	? Question 5 - The statement "something is B and not A" is consistent with the diagram. True or false? [Show answer] True.
	? Question 6 - Is the statement "everything is A or C" is consistent with this diagram? [Show answer] No.
	? Question 7 - Is the statement "if something is B then it is not C" true according to this diagram? [Show answer] Not enough information to decide.