Module: Venn diagrams
Quote of the page
Problems are to the mind what exercise is to the muscles; they toughen and make strong.
- Norman Vincent Peale
So far we have used ticks to cut across only two bounded regions. But of course there are other possibilities:
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 (or both), but which are not A. So what the diagram says is that there is something of this kind.
So what if we just want to represent the fact that something is A? Here is one way to draw the diagram:
Notice that the tick cuts across all the different regions within the A circle, and is completely enclosed by it.
We can now 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, and it 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 the following 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.
a. Is the statement "something is either B or C" true according to the diagram?
b. Is the diagram consistent with the statement "Everything is B or not C, or both"?
c. Is the diagram consistent with the statement "Something is B and not A"?
d. Is the diagram consistent with the statement "Everything is A or C"?
e. What does the diagram tell us?