Euler diagrams consist of simple closed shapes in a two-dimensional plane that each depict a set or category.
A Venn diagram must contain all 2n logically possible zones of overlap between its n curves, representing all combinations of inclusion/exclusion of its constituent sets.
Couturat[4] observed that, in a direct algorithmic (formal, systematic) manner, one cannot derive reduced Boolean equations, nor does it show how to arrive at the conclusion "NO x is z".
They give examples of Venn diagrams to solve example switching-circuit problems, but end up with this statement: In Chapter 6, section 6.4 "Karnaugh map representation of Boolean functions" they begin with: The history of Karnaugh's development of his "chart" or "map" method is obscure.
A Venn diagram must contain all 2n logically possible zones of overlap between its n curves, representing all combinations of inclusion/exclusion of its constituent sets.
There are examples of Euler diagrams with 9 sets that are not drawable using simple closed curves without the creation of unwanted zones since they would have to have non-planar dual graphs.
This example shows the Euler and Venn diagrams and Karnaugh map deriving and verifying the deduction "No Xs are Zs".
If the evaluation of the truth table produces all 1s under the implication-sign (→, the so-called major connective) then P → Q is a tautology.
But given the demonstration that P → Q is tautology, the stage is now set for the use of the procedure of modus ponens to "detach" Q: "No Xs are Zs" and dispense with the terms on the left.
[nb 1] Modus ponens (or "the fundamental rule of inference"[15]) is often written as follows: The two terms on the left, P → Q and P, are called premises (by convention linked by a comma), the symbol ⊢ means "yields" (in the sense of logical deduction), and the term on the right is called the conclusion: For the modus ponens to succeed, both premises P → Q and P must be true.
[nb 2] One is now free to "detach" the conclusion "No Xs are Zs", perhaps to use it in a subsequent deduction (or as a topic of conversation).
Euler diagram illustrating that the set of "animals with four legs" is a subset of "animals", but the set of "minerals" is a disjoint set (it has no members in common) with "animals"
Euler diagram showing the relationships between different
Solar System
objects
A page from Hamilton's
Lectures on Logic;
the symbols
A
,
E
,
I
, and
O
refer to four types of categorical statement which can occur in a
syllogism
(see
descriptions, left
) The small text to the left erroneously says: "The first employment of circular diagrams in logic improperly ascribed to Euler. To be found in Christian Weise", a book which was actually written by Johann Christian Lange.
[
2
]
[
3
]
The diagram to the right is from Couturat
[
4
]
(p 74)
in which he labels the 8 regions of the Venn diagram. The modern name for the "regions" is
minterms
. They are shown in the diagram with the variables
x
,
y
, and
z
per Venn's drawing. The symbolism is as follows: logical
AND
[
&
]
is represented by arithmetic multiplication, and the logical
NOT
[
¬
]
is represented by " ' " after the variable, e.g. the region
x
'
y
'
z
is read as "(
NOT
x
)
AND
(
NOT
y
)
AND
z
" i.e.
(¬
x
) & (¬
y
) &
z
.
Both the Veitch diagram and Karnaugh map show all the
minterms
, but the Veitch is not particularly useful for reduction of formulas. Observe the strong resemblance between the Venn and Karnaugh diagrams; the colors and the variables
x
,
y
, and
z
are per Venn's example.
Composite of two pages from
Venn (1881a)
, pp. 115–116 showing his example of how to convert a syllogism of three parts into his type of diagram; Venn calls the circles "Eulerian circles"
[
10
]
Examples of small
Venn diagrams
(on left)
with shaded regions representing
empty sets
, showing how they can be easily transformed into equivalent Euler diagrams
(right)
Before it can be presented in a Venn diagram or Karnaugh Map, the Euler diagram's syllogism "No
Y
is
Z
, All
X
is
Y
" must first be reworded into the more formal language of the
propositional calculus
: " 'It is not the case that:
Y
AND
Z'
AND 'If an
X
then a
Y'
". Once the propositions are reduced to symbols and a propositional formula ( ~(y & z) & (x → y) ), one can construct the formula's
truth table
; from this table the Venn and/or the Karnaugh map are readily produced. By use of the adjacency of "1"s in the Karnaugh map (indicated by the grey ovals around terms 0 and 1 and around terms 2 and 6) one can "reduce" the example's
Boolean equation
i.e. (x'y'z' + x'y'z) + (x'yz' + xyz') to just two terms: x'y' + yz'. But the means for deducing the notion that "No X is Z", and just how the reduction relates to this deduction, is not forthcoming from this example.
