3

It is said on page 15 Smullyan's A beginner's guide to mathematical logic, that

What do we do if we have more than three unknowns, say, A, B, C, D? Well, we can no longer draw circles, but still the four sets divide I into 16 basics regions, and we can number them in such a way that :

A = (1,2,3,4,5,6,7,8)

B = (1.2.3.4.9.10.11.12)

C = (1,2,5,6,9,10,13,14)

D = (1,3,5,7,9,11,13,15)

Why does he say we can no longer draw circles? I did and there are less regions than he has - I got only seven per circle? This would give only 14 regions total in I.

9

It's true that a Venn diagram for four unknowns can't be drawn with circles. This is essentially a theorem of geometry. Basically, after you draw one with three circles (having eight regions, including the outer region), there is no way to draw a fourth circle that includes parts of all eight regions.

However, there is a fairly standard way to draw a Venn diagram for four unknowns using four ellipses. This was invented by Venn himself:

Venn diagram with four ovals

Note that this diagram has 16 regions, including the outer region.

  • And I wonder how did he get those numbers there, let's say C = (1,2,5,6,9,10,13,14) - as he didn't use diagram, how can one know which numbers exactly will go with each set? @Jim Belk – Aili J. Dec 8 '16 at 14:13
  • 1
    @AiliJ: If you look closely at the numbers in your question, you'll notice that group D includes all the odd numbers, i.e. it starts at 1 and skips every second number. Group C starts with 1 and 2, skips 3 and 4, includes 5 and 6, skip 7 and 8, and so on. Group B starts with the numbers 1-4, skips the next four numbers, includes the next four, and so on. And group A, of course, starts with the lowest 8 numbers out of 16, and skips the following 8. To visualize this, you may want to draw a grid with 16 columns (1-16) and four rows (A-D), and mark the regions from each group in the grid. – Ilmari Karonen Dec 8 '16 at 21:18
  • @IlmariKaronen I found that later in my book, too - there was an example shown with five groups; I wonder how would the sixth group be composed? – Aili J. Dec 8 '16 at 21:27
  • 1
    @AiliJ: The sixth group would include the first 32 numbers, then skip the next 32, then (if there are more than six groups) again include the next 32 numbers, skip the next 32 and so on. You may notice a pattern here. – Ilmari Karonen Dec 8 '16 at 21:31
1

All of the 16 regions might be expressed as boolean expressions:

A ∩ B ∩ C ∩ D
A ∩ B ∩ C ∩ D'
A ∩ B ∩ C' ∩ D
A ∩ B ∩ C' ∩ D'
A ∩ B' ∩ C ∩ D 
A ∩ B' ∩ C ∩ D' 
A ∩ B' ∩ C' ∩ D
A ∩ B' ∩ C' ∩ D'
A' ∩ B ∩ C ∩ D
A' ∩ B ∩ C ∩ D'
A' ∩ B ∩ C' ∩ D
A' ∩ B ∩ C' ∩ D'
A' ∩ B' ∩ C ∩ D 
A' ∩ B' ∩ C ∩ D' 
A' ∩ B' ∩ C' ∩ D
A' ∩ B' ∩ C' ∩ D'

Notice that the last set D has to both intersect and not intersect with all of the previous combinations consisting of the sets A, B and C. For example, given the first combination, D would have to both intersect with it and not intersect with it:

(A ∩ B ∩ C) ∩ D
(A ∩ B ∩ C) ∩ D'

In terms of the diagram, that means you would have to draw a circle that would pass through every subregion of the following diagram:

enter image description here

Imagine an ellipse cutting through the lower half of the diagram so that it cuts across 4, 2, 1, 5 and 7. It could also cut across 6 and include parts of 8. However, that still leaves the section 3 which didn't get divided:

enter image description here

As Jim Belk pointed out, it's not possible with circles, but it is with ellipses. (And no, I didn't know that when I first posted this answer.)

  • Just found four circles drawn as I did with 13 regions in Wikipedia (it is called Euler diagram)??? – Aili J. Dec 8 '16 at 14:06
  • and those Boolean expressions are really clever way to use here, didn't even know one can count those sets like that! Thank You – Aili J. Dec 8 '16 at 14:24
  • @AiliJ. To answer your question above, each of the expressions corresponds to one of the numbers the author uses. You can discover which one by writing the expression without a negation for every set in which it occurs. For example, 2 shows up in the sets for A, B and C. That means those will not be negated and D will be negated: A ∩ B ∩ C ∩ D'. – user3017 Dec 8 '16 at 14:35
  • But how do you know in the first place, that 2 shows up in the sets for A,B, and C? @ Pé de Leão – Aili J. Dec 8 '16 at 14:47
  • @AiliJ. I'm just looking at the sets you posted in your question. Another example: 3 shows up for sets A, B and D, so the expression would be: A ∩ B ∩ C' ∩ D. – user3017 Dec 8 '16 at 14:55

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