Lihat sumber Diagram Euler

{{terjemah|Inggris}}
{{about|Lingkaran Euler dari teori dan logika himpunan|lingkaran geometri Euler|Lingkaran sembilan titik}}
[[Berkas:EulerDiagram.svg|thumb|Diagram Euler yang menggambarkan bahwa himpunan "hewan berkaki empat" adalah himpunan bagian dari "hewan", tetapi himpunan "mineral" adalah terputus-putus (tidak memiliki anggota yang sama) dengan "hewan"]]
[[Berkas:Euler_diagram_of_solar_system_bodies.svg|thumb|right|300px|Diagram Euler yang menunjukkan hubungan antara objek [[Tata Surya]] yang berbeda]]

'''Diagram Euler''' ({{IPAc-en|ˈ|ɔɪ|l|ər}},) adalah [[diagram]] cara metik untuk mewakili [[Himpunan (matematika)|himpunan]] dan hubungannya. Mereka sangat berguna untuk menjelaskan hierarki yang kompleks dan definisi yang tumpang tindih. Mereka mirip dengan teknik diagram himpunan lainnya, [[diagram Venn]]. Tidak seperti diagram Venn, yang menunjukkan semua kemungkinan hubungan antara set yang berbeda, diagram Euler hanya menampilkan relasi yang relevan.

Penggunaan pertama "lingkaran Eulerian" biasanya dikaitkan dengan ahli matematika Swiss [[Leonhard Euler]] (1707–1783). Di Amerika Serikat, diagram Venn dan Euler digabungkan sebagai bagian dari instruksi di [[teori himpunan]] sebagai bagian dari gerakan [[matematika baru]] tahun 1960-an. Sejak saat itu, mereka juga diadopsi oleh bidang kurikulum lain seperti membaca<ref>{{Cite web |url=http://www.readingquest.org/strat/venn.html |title=Strategies for Reading Comprehension Venn Diagrams |access-date=2009-06-20 |archive-url=https://web.archive.org/web/20090429093334/http://readingquest.org/strat/venn.html |archive-date=2009-04-29 |url-status=dead }}</ref> serta organisasi dan bisnis.

Diagram Euler terdiri dari bentuk tertutup sederhana dalam bidang dua dimensi yang masing-masing menggambarkan himpunan atau kategori. Bagaimana atau jika bentuk-bentuk ini tumpang tindih mendemonstrasikan hubungan antar set. Setiap kurva membagi bidang menjadi dua wilayah atau "zona": interior, yang secara simbolis mewakili [[elemen (matematika)|elemen]] dari himpunan, dan eksterior, yang mewakili semua elemen yang bukan anggota himpunan. Kurva yang tidak tumpang tindih mewakili [[set terputus-putus]], yang tidak memiliki elemen yang sama. Dua kurva yang tumpang tindih mewakili himpunan yang [[perpotongan (teori himpunan)|berpotongan]], yang memiliki elemen yang sama; zona di dalam kedua kurva mewakili himpunan elemen yang sama untuk kedua himpunan ([[perpotongan (teori himpunan)|perpotongan]] dari himpunan). Sebuah kurva yang seluruhnya berada di dalam interior lainnya adalah [[subhimpunan]] darinya.

[[Diagram Venn]] adalah bentuk diagram Euler yang lebih terbatas. Diagram Venn harus berisi semua 2<sup>''n''</sup> secara logis kemungkinan zona tumpang tindih antara kurva '' n '', mewakili semua kombinasi penyertaan/pengecualian himpunan konstituennya. Region bukan bagian dari himpunan ditunjukkan dengan mewarnai mereka dengan warna hitam, berbeda dengan diagram Euler, di mana keanggotaan dalam himpunan ditunjukkan dengan tumpang tindih serta warna.

== Sejarah ==
{{Dalam perbaikan|bagian ini jangan dihapus}}
[[Berkas:Hamilton Lectures on Logic 1874 Euler Diagrams.png|thumb|344x344px|Sebuah halaman dari '' Lectures on Logic '' Hamilton. Simbolisme A, E, I, dan O mengacu pada pernyataan kategoris yang dapat terjadi dalam [[silogisme]]. Teks kecil di sebelah kiri secara keliru menyatakan: "Penggunaan pertama diagram lingkaran dalam logika yang secara tidak tepat dianggap berasal dari Euler. Dapat ditemukan di Christian Weise ", sebuah buku yang sebenarnya ditulis oleh Johann Christian Lange.<ref name="Venn_1881"/><ref name="Gailand_1967"/>]]
[[Berkas:Couturat 1914 and Venn assignments1.jpg|thumb|500px|right|Di sebelah kanan adalah halaman 74 dari Couturat 1914 di mana dia memberi label 8 wilayah dari diagram Venn. Nama modern untuk "wilayah" ini adalah [[minterm]]. Ini ditunjukkan di sebelah kiri dengan variabel x, y, dan z per gambar Venn. Simbolismenya adalah sebagai berikut: logika AND (&) diwakili oleh perkalian aritmatika, dan logika NOT (~) diwakili oleh "'" setelah variabel, mis. daerah x'y'z dibaca sebagai "BUKAN x DAN BUKAN y DAN z" yaitu ~ x & ~]]

[[Berkas:Veitch and Karnaugh 3.jpg|thumb|300px|right|Diagram Veitch dan peta Karnaugh menunjukkan semua [[minterms]], tetapi Veitch tidak terlalu berguna untuk reduksi rumus. Amati kemiripan yang kuat antara diagram Venn dan Karnaugh; warna dan variabel x, y, dan z sesuai dengan contoh Venn.]]

Seperti yang ditunjukkan pada ilustrasi di sebelah kanan, [[Sir William Hamilton, Baronet ke-9|Sir William Hamilton]] dalam bukunya yang diterbitkan secara anumerta '' Lectures on Metaphysics and Logic '' (1858–60) secara keliru menegaskan ... abstraksi Logika "(hlm. 180) bukanlah [[Leonhard Paul Euler]] (1707–1783) melainkan [[Christian Weise]] (1642–1708) dalam bukunya '' Nucleus Logicae Weisianae '' yang muncul pada 1712 secara anumerta, bagaimanapun, buku terakhir sebenarnya ditulis oleh Johann Christian Lange daripada Weise.<ref name="Venn_1881"/><ref name="Gailand_1967"/> He references Euler's ''[[Letters to a German Princess]]'' [Partie II, Lettre XXXV, February 17, 1791, ed. Cournot (1842), pp. 412-417.  – ED.]<ref group="nb">Pada saat kuliah Hamilton ini diterbitkan, Hamilton juga telah meninggal. Editornya (disimbolkan dengan ED.), bertanggung jawab untuk sebagian besar catatan kaki, adalah para ahli logika [[Henry Longueville Mansel]] dan [[John Veitch (penyair)|John Veitch]].</ref>

Dalam ilustrasi Hamilton, empat [[proposisi kategorikal]] yang dapat terjadi dalam [[silogisme]] seperti yang disimbolkan oleh gambar A, E, I dan O adalah:<ref>Hamilton 1860:179. The examples are from Jevons 1881:71ff.</ref>
* J: The '' Universal Affirmative '', Contoh: "Semua logam adalah unsur".
* E: The '' Universal Negative '', Contoh: "Tidak ada logam yang merupakan zat majemuk".
* I: The '' Particular Affirmative '', Contoh: "Beberapa logam rapuh".
* O: The '' Particular Negative '', Contoh: "Beberapa logam tidak rapuh".

Dalam karyanya tahun 1881 '' Symbolic Logic '' Bab V "Diagrammatic Representation", [[John Venn]] (1834–1923) mengomentari prevalensi yang luar biasa dari diagram Euler:
: "...dari enam puluh risalah logis pertama, yang diterbitkan selama sekitar abad terakhir, yang dikonsultasikan untuk tujuan ini: -beberapa secara acak, karena kebetulan mereka paling mudah diakses: -tampaknya tiga puluh empat meminta bantuan diagram, hampir semua ini memanfaatkan Skema Eulerian. "(Catatan kaki 1 halaman 100)

[[Berkas:Venn 1881 p 115-116 pasteup.jpg|thumb|300px|left|Gabungan dari dua halaman 115–116 dari Venn 1881 menunjukkan contoh bagaimana mengubah silogisme tiga bagian menjadi tipe diagramnya. Venn menyebut lingkaran tersebut "lingkaran Eulerian" (cf Sandifer 2003, Venn 1881: 114 dll) dalam "skema Eulerian" (Venn 1881: 100) dari "diagram Euler kuno" (Venn 1881: 113).]]

Namun demikian, ia berpendapat, "ketidakmampuan skema ini untuk tujuan Logika yang sangat umum" (halaman 100), "Ini cocok tetapi buruk bahkan dengan empat proposisi dari Logika umum yang biasanya diterapkan." Venn mengakhiri babnya dengan observasi yang diilustrasikan dalam contoh di bawah — bahwa penggunaannya didasarkan pada praktik dan intuisi, bukan pada praktik [[algoritme]] yang ketat:
:“Nyatanya ... diagram-diagram tersebut tidak hanya tidak cocok dengan skema proposisi biasa yang digunakan untuk menggambarkannya, tetapi tampaknya tidak memiliki skema proposisi yang diakui sehingga mereka dapat secara konsisten berafiliasi. " (hlm. 124–125)
<!--
Akhirnya, dalam Bab XX CATATAN SEJARAH Venn mendapat kritik penting (dicetak miring dalam kutipan di bawah); amati dalam ilustrasi Hamilton bahwa O ('' Particular Negative '') dan I ('' Particular Affirmative '') hanya diputar:

:"Sekarang kita sampai pada lingkaran terkenal Euler yang pertama kali dijelaskan dalam bukunya '' Lettres a une Princesse d'Allemagne '' (Surat 102-105). Titik lemah tentang ini terdiri dari fakta bahwa mereka hanya menggambarkan secara ketat hubungan kelas yang sebenarnya dengan yang lain, rather than the imperfect knowledge of these relations which we may possess, or wish to convey, by means of the proposition. Accordingly they will not fit in with the propositions of common logic, but demand the constitution of a new group of appropriate elementary propositions.... This defect must have been noticed from the first ''in the case of the particular affirmative and negative, for the same diagram is commonly employed to stand for them both, which it does indifferently well''". (italics added: page 424)

(Sandifer 2003 reports that Euler makes such observations too; Euler reports that his figure 45 (a simple intersection of two circles) has 4 different interpretations). Whatever the case, armed with these observations and criticisms, Venn then demonstrates (pp.&nbsp;100–125) how he derived what has become known as his [[Venn diagrams]] from the "...old-fashioned Euler diagrams." In particular he gives an example, shown on the left.

By 1914, [[Louis Couturat]] (1868–1914) had labeled the terms as shown on the drawing on the right. Moreover, he had labeled the ''exterior region'' (shown as a'b'c') as well. He succinctly explains how to use the diagram – one must ''strike out'' the regions that are to vanish:
:"VENN'S method is translated in geometrical diagrams which represent all the constituents, so that, in order to obtain the result, we need only ''strike out (by shading)'' those which are made to vanish by the data of the problem." (italics added p. 73)

Given the Venn's assignments, then, the unshaded areas ''inside'' the circles can be summed to yield the following equation for Venn's example:
: "No Y is Z and ALL X is Y: therefore No X is Z" has the equation x'yz' + xyz' + x'y'z for the unshaded area ''inside'' the circles (but this is not entirely correct; see the next paragraph).

In Venn the 0th term, x'y'z', i.e. the background surrounding the circles, does not appear.  Nowhere is it discussed or labeled, but Couturat corrects this in his drawing. The correct equation must include this unshaded area shown in boldface:
: "No Y is Z and ALL X is Y: therefore No X is Z" has the equation x'yz' + xyz' + x'y'z + ''' x'y'z' '''.

In modern usage the Venn diagram includes a "box" that surrounds all the circles; this is called the universe of discourse or the [[domain of discourse]].

Couturat now observes that, in a direct [[algorithm]]ic (formal, systematic) manner, one cannot derive reduced Boolean equations, nor does it show how to arrive at the conclusion "No X is Z". Couturat concluded that the process "has ... serious inconveniences as a method for solving logical problems":
:"It does not show how the data are exhibited by canceling certain constituents, nor does it show how to combine the remaining constituents so as to obtain the consequences sought. In short, it serves only to exhibit one single step in the argument, namely the equation of the problem; it dispenses neither with the previous steps, i. e., "throwing of the problem into an equation" and the transformation of the premises, nor with the subsequent steps, i. e., the combinations that lead to the various consequences. Hence it is of very little use, inasmuch as the constituents can be represented by algebraic symbols quite as well as by plane regions, and are much easier to deal with in this form."(p. 75)

Thus the matter would rest until 1952 when [[Maurice Karnaugh]] (1924–&nbsp;) would adapt and expand a method proposed by [[Edward W. Veitch]]; this work would rely on the [[truth table]] method precisely defined in [[Emil Post]]'s 1921 PhD thesis "Introduction to a general theory of elementary propositions" and the application of propositional logic to [[switching logic]] by (among others) [[Claude Shannon]], [[George Stibitz]], and [[Alan Turing]].<ref group="nb">See footnote at [[George Stibitz]].</ref> For example, in chapter "Boolean Algebra", Hill and Peterson (1968, 1964) present sections 4.5ff "Set Theory as an Example of Boolean Algebra", and in it they present the Venn diagram with shading and all. They give examples of Venn diagrams to solve example switching-circuit problems, but end up with this statement:
::"For more than three variables, the basic illustrative form of the Venn diagram is inadequate. Extensions are possible, however, the most convenient of which is the Karnaugh map, to be discussed in Chapter 6." (p. 64)
In Chapter 6, section 6.4 "Karnaugh Map Representation of Boolean Functions" they begin with:
::"The Karnaugh map<sup>1</sup> [<sup>1</sup>Karnaugh 1953] is one of the most powerful tools in the repertory of the logic designer. ... A Karnaugh map may be regarded either as a pictorial form of a truth table or as an extension of the Venn diagram." (pp. 103–104)

The history of Karnaugh's development of his "chart" or "map" method is obscure. Karnaugh in his 1953 referenced Veitch 1951, Veitch referenced [[Claude E. Shannon]] 1938 (essentially Shannon's Master's thesis at [[M.I.T.]]), and Shannon in turn referenced, among other authors of logic texts, Couturat 1914. In Veitch's method the variables are arranged in a rectangle or square; as described in [[Karnaugh map]], Karnaugh in his method changed the order of the variables to correspond to what has become known as (the vertices of) a [[hypercube]].
-->

== Hubungan antara diagram Euler dan Venn ==

- Proses pengembangan -
<!-- sembunyikan dulu
[[Berkas:Syllogism-Set-Diagrams.svg|thumb|300px|right|Contoh [[diagram Venn]] kecil '' (di kiri) '' dengan daerah teduh mewakili [[set kosong]], menunjukkan bagaimana mereka dapat dengan mudah diubah menjadi diagram Euler yang setara '' (kanan) '']]
[[Venn diagram]]s are a more restrictive form of Euler diagrams.  A Venn diagram must contain all 2<sup>''n''</sup> logically possible zones of overlap between its ''n'' curves, representing all combinations of inclusion/exclusion of its constituent sets.  Regions not part of the set are indicated by coloring them black, in contrast to Euler diagrams, where membership in the set is indicated by overlap as well as color.  When the number of sets grows beyond 3 a Venn diagram becomes visually complex, especially compared to the corresponding Euler diagram. The difference between Euler and Venn diagrams can be seen in the following example. Take the three sets:

* <math>A = \{1,\, 2,\, 5\}</math>
* <math>B = \{1,\, 6\}</math>
* <math>C = \{4,\, 7\}</math>

The Euler and the Venn diagrams of those sets are:

<gallery widths="300">
File:3-set Euler diagram.svg|Euler diagram
File:3-set Venn diagram.svg|Venn diagram
</gallery>

In a logical setting, one can use model theoretic semantics to interpret Euler diagrams, within a [[universe of discourse]]. In the examples below, the Euler diagram depicts that the sets ''Animal'' and ''Mineral'' are disjoint since the corresponding curves are disjoint, and also that the set ''Four Legs'' is a subset of the set of ''Animal''s. The Venn diagram, which uses the same categories of ''Animal'', ''Mineral'', and ''Four Legs'', does not encapsulate these relationships. Traditionally the ''emptiness'' of a set in Venn diagrams is depicted by shading in the region. Euler diagrams represent ''emptiness'' either by shading or by the absence of a region.

Often a set of well-formedness conditions are imposed; these are topological or geometric constraints imposed on the structure of the diagram.  For example, connectedness of zones might be enforced, or concurrency of curves or multiple points might be banned, as might tangential intersection of curves. In the adjacent diagram, examples of small Venn diagrams are transformed into Euler diagrams by sequences of transformations; some of the intermediate diagrams have concurrency of curves. However, this sort of transformation of a Venn diagram with shading into an Euler diagram without shading is not always possible. 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.

=== Example: Euler- to Venn-diagram and Karnaugh map ===
This example shows the Euler and Venn diagrams and Karnaugh map deriving and verifying the deduction "No ''X''s are ''Z''s".
In the illustration and table the following logical symbols are used:
* 1 can be read as "true", 0 as "false"
* ~ for NOT and abbreviated to ' when illustrating the minterms e.g. x' =<sub>defined</sub> NOT x,
* + for Boolean OR (from [[Boolean algebra (logic)|Boolean algebra]]: 0+0=0, 0+1 = 1+0 = 1, 1+1=1)
* & (logical AND) between propositions; in the mintems AND is omitted in a manner similar to arithmetic multiplication: e.g. x'y'z =<sub>defined</sub> ~x & ~y & z (From Boolean algebra: 0*0=0, 0*1 = 1*0=0, 1*1 = 1, where * is shown for clarity)
* → (logical IMPLICATION): read as IF ... THEN ..., or " IMPLIES ", ''P'' → ''Q'' =<sub>defined</sub> NOT ''P'' OR ''Q''

[[File:Veitch and Karnaugh truth table 4.jpg|thumb|921px|center|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: {{var|Y}} AND {{var|Z'}} AND 'If an {{var|X}} then a {{var|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.]]

Given a proposed conclusion such as "No ''X'' is a ''Z''", one can test whether or not it is a correct [[Deductive reasoning|deduction]] by use of a [[truth table]]. The easiest method is put the starting formula on the left (abbreviate it as ''P'') and put the (possible) deduction on the right (abbreviate it as ''Q'') and connect the two with [[logical implication]] i.e. ''P'' → ''Q'', read as IF ''P'' THEN ''Q''. 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 (logic)|tautology]]. Given this fact, one can "detach" the formula on the right (abbreviated as ''Q'') in the manner described below the truth table.

Given the example above, the formula for the Euler and Venn diagrams is:
: "No ''Y''s are ''Z''s" and "All ''X''s are ''Y''s": ( ~(y & z) & (x → y) ) =<sub>defined</sub> ''P''
And the proposed deduction is:
: "No ''X''s are ''Z''s": ( ~ (x & z) ) =<sub>defined</sub> ''Q''

So now the formula to be evaluated can be abbreviated to:
: ( ~(y & z) & (x → y) ) → ( ~ (x & z) ): ''P'' → ''Q''
: IF ( "No ''Y''s are ''Z''s" and "All ''X''s are ''Y''s" ) THEN ( "No ''X''s are ''Z''s" )

{| style="width:auto; border:1px solid darkgray; border-collapse: collapse; margin-left: auto; margin-right: auto; text-align:center"
|+ The Truth Table demonstrates that the formula ( ~(y & z) & (x → y) ) → ( ~ (x & z) ) is a tautology as shown by all 1s in yellow column.
|-
! style="width:80pt" | Square #
! style="width:160pt"| Venn, Karnaugh region
! style="width:13pt" |
! style="width:20pt" | x
! style="width:20pt" | y
! style="width:20pt" | z
! style="width:13pt" |
! style="width:30pt" | (~
! style="width:30pt" | (y
! style="width:30pt" | &
! style="width:30pt" | z)
! style="width:30pt" | &
! style="width:30pt" | (x
! style="width:30pt" | →
! style="width:30pt" | y))
! style="width:30pt" | →
! style="width:30pt" | (~
! style="width:30pt" | (x
! style="width:30pt" | &
! style="width:30pt" | z))
|- style="font-size:12pt; height:25pt"
| {{EquationRef|0}}
| x'y'z'
| &nbsp;
| 0
| 0
| 0
| &nbsp;
| 1
| 0
| 0
| 0
! 1
| 0
| 1
| 0
|style="background-color:#F5FF93" | 1
! 1
| 0
| 0
| 0
|- style="font-size:12pt; height:25pt"
| {{EquationRef|1}}
| x'y'z
| &nbsp;
| 0
| 0
| 1
| &nbsp;
| 1
| 0
| 0
| 1
! 1
| 0
| 1
| 0
|style="background-color:#F5FF93"  | 1
! 1
| 0
| 0
| 1
|- style="font-size:12pt; height:25pt"
| {{EquationRef|2}}
| x'yz'
| &nbsp;
| 0
| 1
| 0
| &nbsp;
| 1
| 1
| 0
| 0
! 1
| 0
| 1
| 1
|style="background-color:#F5FF93"  | 1
! 1
| 0
| 0
| 0
|- style="font-size:12pt; height:25pt"
|style="background-color:#EDB9B9"  | {{EquationRef|3}}
|style="background-color:#EDB9B9"  | x'yz
| &nbsp;
|style="background-color:#EDB9B9"  | 0
|style="background-color:#EDB9B9"  | 1
|style="background-color:#EDB9B9"  | 1
| &nbsp;
|style="background-color:#EDB9B9"  | 0
|style="background-color:#EDB9B9"  | 1
|style="background-color:#EDB9B9"  | 1
|style="background-color:#EDB9B9"  | 1
!style="background-color:#EDB9B9;" | 0
|style="background-color:#EDB9B9"  | 0
|style="background-color:#EDB9B9"  | 1
|style="background-color:#EDB9B9"  | 1
|style="background-color:#F5FF93"  | 1
!style="background-color:#EDB9B9;" | 1
|style="background-color:#EDB9B9"  | 0
|style="background-color:#EDB9B9"  | 0
|style="background-color:#EDB9B9"  | 1
|- style="font-size:12pt; height:25pt"
|style="background-color:#DBE5F1"  | {{EquationRef|4}}
|style="background-color:#DBE5F1"  | xy'z'
| &nbsp;
|style="background-color:#DBE5F1"  | 1
|style="background-color:#DBE5F1"  | 0
|style="background-color:#DBE5F1"  | 0
| &nbsp;
|style="background-color:#DBE5F1"  | 1
|style="background-color:#DBE5F1"  | 0
|style="background-color:#DBE5F1"  | 0
|style="background-color:#DBE5F1"  | 0
!style="background-color:#DBE5F1;" | 0
|style="background-color:#DBE5F1"  | 1
|style="background-color:#DBE5F1"  | 0
|style="background-color:#DBE5F1"  | 0
|style="background-color:#F5FF93"  | 1
!style="background-color:#DBE5F1;" | 1
|style="background-color:#DBE5F1"  | 1
|style="background-color:#DBE5F1"  | 0
|style="background-color:#DBE5F1"  | 0
|- style="font-size:12pt; height:25pt"
|style="background-color:#FFCCFF"  | {{EquationRef|5}}
|style="background-color:#FFCCFF"  | xy'z
| &nbsp;
|style="background-color:#FFCCFF"  | 1
|style="background-color:#FFCCFF"  | 0
|style="background-color:#FFCCFF"  | 1
| &nbsp;
|style="background-color:#FFCCFF"  | 1
|style="background-color:#FFCCFF"  | 0
|style="background-color:#FFCCFF"  | 0
|style="background-color:#FFCCFF"  | 1
!style="background-color:#FFCCFF;" | 0
|style="background-color:#FFCCFF"  | 1
|style="background-color:#FFCCFF"  | 0
|style="background-color:#FFCCFF"  | 0
|style="background-color:#F5FF93"  | 1
!style="background-color:#FFCCFF;" | 0
|style="background-color:#FFCCFF"  | 1
|style="background-color:#FFCCFF"  | 1
|style="background-color:#FFCCFF"  | 1
|- style="font-size:12pt; height:25pt"
| {{EquationRef|6}}
| xyz'
| &nbsp;
| 1
| 1
| 0
| &nbsp;
| 1
| 1
| 0
| 0
! 1
| 1
| 1
| 1
|style="background-color:#F5FF93"  | 1
! 1
| 1
| 0
| 0
|- style="font-size:12pt; height:25pt"
|style="background-color:#D6B4D0"  | {{EquationRef|7}}
|style="background-color:#D6B4D0"  | xyz
| &nbsp;
|style="background-color:#D6B4D0"  | 1
|style="background-color:#D6B4D0"  | 1
|style="background-color:#D6B4D0"  | 1
| &nbsp;
|style="background-color:#D6B4D0"  | 0
|style="background-color:#D6B4D0"  | 1
|style="background-color:#D6B4D0"  | 1
|style="background-color:#D6B4D0"  | 1
!style="background-color:#D6B4D0;" | 0
|style="background-color:#D6B4D0"  | 1
|style="background-color:#D6B4D0"  | 1
|style="background-color:#D6B4D0"  | 1
|style="background-color:#F5FF93"  | 1
!style="background-color:#D6B4D0;" | 0
|style="background-color:#D6B4D0"  | 1
|style="background-color:#D6B4D0"  | 1
|style="background-color:#D6B4D0"  | 1
|}

At this point the above implication ''P'' → ''Q'' (i.e. ~(y & z) & (x → y) ) →  ~(x & z) ) is still a formula, and the deduction – the "detachment" of ''Q'' out of ''P'' → ''Q'' – has not occurred. 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 ''X''s are ''Z''s" and dispense with the terms on the left.<ref group="nb">This is a sophisticated concept. Russell and Whitehead (2nd edition 1927) in their ''[[Principia Mathematica]]'' describe it this way: "The trust in inference is the belief that if the two former assertions [the premises P, P→Q ] are not in error, the final assertion is not in error . . . An inference is the dropping of a true premiss [sic]; it is the dissolution of an implication" (p. 9). Further discussion of this appears in "Primitive Ideas and Propositions" as the first of their "primitive propositions" (axioms): *1.1 Anything implied by a true elementary proposition is true" (p. 94). In a footnote the authors refer the reader back to Russell's 1903 ''Principles of Mathematics'' §38.</ref>

''Modus ponens'' (or "the fundamental rule of inference"<ref>cf Reichenbach 1947:64</ref>) 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'':
: ''P'' → ''Q'', ''P'' ⊢ ''Q''

For the modus ponens to succeed, both premises P → Q and P must be ''true''. Because, as demonstrated above the premise ''P'' → ''Q'' is a tautology, "truth" is always the case no matter how x, y and z are valued, but "truth" is only the case for ''P'' in those circumstances when ''P'' evaluates as "true" (e.g. rows {{EquationNote|0}} OR {{EquationNote|1}} OR {{EquationNote|2}} OR {{EquationNote|6}}: x'y'z' + x'y'z + x'yz' + xyz' = x'y' + yz').<ref group="nb">Reichenbach discusses the fact that the implication ''P'' → ''Q'' need not be a tautology (a so-called "tautological implication"). Even "simple" implication (connective or adjunctive) work, but only for those rows of the truth table that evaluate as true, cf Reichenbach 1947:64–66.</ref>
: ''P'' → ''Q'' , ''P'' ⊢ ''Q''
:* i.e.: ( ~(y & z) & (x → y) ) → ( ~ (x & z) ) , ( ~(y & z) & (x → y) ) ⊢ ( ~ (x & z) )
:* i.e.: IF "No ''Y''s are ''Z''s" and "All ''X''s are ''Y''s" ''THEN'' "No ''X''s are ''Z''s", "No ''Y''s are ''Z''s" and "All ''X''s are ''Y''s" ⊢ "No ''X''s are ''Z''s"

One is now free to "detach" the conclusion "No ''X''s are ''Z''s", perhaps to use it in a subsequent deduction (or as a topic of conversation).

The use of tautological implication means that other possible deductions exist besides "No ''X''s are ''Z''s"; the criterion for a successful deduction is that the 1s under the sub-major connective on the right ''include'' all the 1s under the sub-major connective on the left (the ''major'' connective being the implication that results in the tautology). For example, in the truth table, on the right side of the implication (→, the major connective symbol) the bold-face column under the sub-major connective symbol " '''~''' " has the all the same 1s that appear in the bold-faced column under the left-side sub-major connective '''&''' (rows {{EquationNote|0}}, {{EquationNote|1}}, {{EquationNote|2}} and {{EquationNote|6}}), plus two more (rows {{EquationNote|3}} and {{EquationNote|4}}).-->

== Galeri ==
<gallery>
Berkas:VennDiagram.svg|[[Diagram Venn]] menunjukkan semua kemungkinan persimpangan.
Berkas:Supranational European Bodies.svg|Diagram Euler memvisualisasikan situasi nyata, hubungan antara berbagai [[Organisasi internasional di Eropa|organisasi supranasional Eropa]]. ([[Templat:Supranational European Bodies|versi yang dapat diklik]])
Berkas:Euler and Venn diagrams.svg|Diagram lucu membandingkan Euler dan [[diagram Venn]].
Berkas:Euler diagram of triangle types.svg|Diagram euler dari jenis [[segitiga]], menggunakan definisi bahwa segitiga sama kaki memiliki setidaknya (bukan persis) 2 sisi yang sama.
Berkas:British Isles Euler diagram 15.svg|Diagram Euler dari terminologi [[Kepulauan Inggris]].

Berkas:Venn and Euler diagrams of 3-ary Boolean relations.svg|22 (dari 256) diagram Venn yang pada dasarnya berbeda dengan 3 lingkaran '' (atas) '' dan diagram Euler yang sesuai '' (bawah) ''<small><br/>Beberapa diagram Euler tidak tipikal, dan beberapa bahkan setara dengan diagram Venn. Area diarsir untuk menunjukkan bahwa mereka tidak mengandung elemen.</small>
</gallery>
{{Badan Eropa Supranasional|size=350px|align=right}}

== Lihat pula ==
* [[Diagram Spider]] - perpanjangan dari diagram Euler yang menambahkan keberadaan ke persimpangan kontur.

== Catatan ==
{{Reflist|group="nb"}}

== Referensi ==
{{Reflist|refs=
<ref name="Venn_1881">{{cite book |author-first=John |author-last=Venn |author-link=John Venn |date=1881 |title=Symbolic Logic |url=https://archive.org/details/symboliclogic00vennuoft |publisher=[[MacMillan and Co.]] |location=London |page=509}}</ref>
<ref name="Gailand_1967">{{cite book |title=The Logic Diagram |author-first=Gailand |author-last=Mac Queen |date=October 1967 |type=Thesis |publisher=[[McMaster University]] |page=5 |url=https://macsphere.mcmaster.ca/bitstream/11375/10794/1/fulltext.pdf |access-date=2017-04-14 |url-status=dead |archive-url=https://web.archive.org/web/20170414163921/https://macsphere.mcmaster.ca/bitstream/11375/10794/1/fulltext.pdf |archive-date=2017-04-14}} (NB. Has a detailed history of the evolution of logic diagrams including but not limited to the Euler diagram.)</ref>
}}

== Bacaan lebih lanjut ==
Menurut tanggal penerbitan:

* [[Sir William Hamilton, 9th Baronet|Sir William Hamilton]] 1860 ''Lectures on Metaphysics and Logic'' edited by [[Henry Longueville Mansel]] and [[John Veitch (poet)|John Veitch]], [[William Blackwood and Sons]], Edinburgh and London.
* [[W. Stanley Jevons]] 1880 ''Elementary Lessons in Logic: Deductive and Inductive. With Copious Questions and Examples, and a Vocabulary of Logical Terms'', [[M. A. MacMillan and Co.]], London and New York.
* [[Alfred North Whitehead]] and [[Bertrand Russell]] 1913 1st edition, 1927 2nd edition ''Principia Mathematica to *56'' Cambridge At The [[Cambridge University Press|University Press]] (1962 edition), UK, no ISBN.
* [[Louis Couturat]] 1914 ''The Algebra of Logic: Authorized English Translation by Lydia Gillingham Robinson with a Preface by Philip E. B. Jourdain'', [[The Open Court Publishing Company]], Chicago and London.
* [[Emil Post]] 1921 "Introduction to a general theory of elementary propositions" reprinted with commentary by [[Jean van Heijenoort]] in Jean van Heijenoort, editor 1967 ''From Frege to Gödel: A Source Book of Mathematical Logic, 1879–1931'', [[Harvard University Press]], Cambridge, MA, {{ISBN|0-674-32449-8}} (pbk.)
* [[Claude E. Shannon]] 1938 "A Symbolic Analysis of Relay and Switching Circuits", ''Transactions American Institute of Electrical Engineers'' vol 57, pp.&nbsp;471–495. Derived from ''Claude Elwood Shannon: Collected Papers'' edited by N.J.A. Solane and Aaron D. Wyner, [[IEEE Press]], New York.
* [[Hans Reichenbach]] 1947 ''Elements of Symbolic Logic'' republished 1980 by [[Dover Publications, Inc.]], NY, {{ISBN|0-486-24004-5}}.
* {{cite journal |author-last=Veitch |author-first=Edward Westbrook |author-link=Edward W. Veitch |title=A Chart Method for Simplifying Truth Functions |journal=Transactions of the 1952 ACM Annual Meeting |series=ACM Annual Conference/Annual Meeting: Proceedings of the 1952 ACM Annual Meeting (Pittsburgh, Pennsylvania, USA) |location=New York, USA |publisher=[[Association for Computing Machinery]] (ACM) |pages=127–133 |date=1952-05-03 |orig-year=1952-05-02 |doi=10.1145/609784.609801}}
* {{cite journal |author-last=Karnaugh |author-first=Maurice |author-link=Maurice Karnaugh |title=The Map Method for Synthesis of Combinational Logic Circuits |journal=[[Transactions of the American Institute of Electrical Engineers, Part I: Communication and Electronics]] |volume=72 |issue=5 |pages=593–599 |date=November 1953 |orig-year=1953-04-23, 1953-03-17 |id=Paper 53-217 |doi=10.1109/TCE.1953.6371932 |url=http://philectrosophy.com/documents/The%20Map%20Method%20For%20Synthesis%20of.pdf |access-date=2017-04-16 |url-status=dead |archive-url=https://web.archive.org/web/20170416232229/http://philectrosophy.com/documents/The%20Map%20Method%20For%20Synthesis%20of.pdf |archive-date=2017-04-16}}
* Frederich J. Hill and Gerald R. Peterson 1968, 1974 ''Introduction to Switching Theory and Logical Design'', [[John Wiley & Sons]], NY, {{ISBN|978-0-471-39882-0}}.
* {{cite web |author-first=Ed |author-last=Sandifer |date=January 2004 |title=How Euler Did It |url=http://www.maa.org/editorial/euler/How%20Euler%20Did%20It%2003%20Venn%20Diagrams.pdf |archive-url=https://web.archive.org/web/20130126165922/http://maa.org/editorial/euler/How%20Euler%20Did%20It%2003%20Venn%20Diagrams.pdf |archive-date=2013-01-26 |website=maa.org}}

== Pranala luar ==
{{commons category|Euler diagrams}}
* Euler Diagrams. Brighton, UK (2004).[http://www.cs.kent.ac.uk/events/conf/2004/euler/eulerdiagrams.html What are Euler Diagrams?]

{{DEFAULTSORT:Diagram Euler}}
[[Kategori:Konsep grafis dalam teori himpunan]]
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