QEDとは 意味・読み方・使い方

語源

From Late Latin QED, from Latin quod erat demonstrandum.^[1]

発音

• (Received 発音) IPA^(key): /ˌkjuːiːˈdiː/
• (General American) IPA^(key): /ˌkjuˌiˈdi/

成句

Q.E.D.

1. (mathematics, dated) Initialism of quod erat demonstrandum (“what had to be proved; what was to be demonstrated”): placed at the end of a mathematical proof to show that the theorem under discussion is proved.
   • 1684 August 30, Mr. Ash, “A New and Easy Way of Demonstrating Some Propositions in Euclid”, in Philosophical Transactions: Giving Some Accompt of the Present Undertakings, Studies and Labours of the Ingenious in Many Considerable Parts of the World, volume XIV, number 162, London: Printed by T. R. for John Martyn, printer to the Royal Society; […] , OCLC 630046584, page 674:

     [A]fter the same manner S and U are proved to be equal, therefore the square of CB is equal to the square of the 2 other sides Q E D.

   • 1803, “The Mathematical Repository, Nº 63”, in The Gentleman’s Diary, or The Mathematical Repository; an Almanack for the Year of Our Lord 1803: […], London: Printed for the Company of Stationers by Nichols and Son, […]; [a]nd sold by George Greenhill, […], OCLC 225667331, page 33:

     Now let any right line meet four harmonicals in ${\displaystyle A}$, ${\displaystyle C}$, ${\displaystyle B}$, ${\displaystyle D}$, and if those harmonicals be parallel, the thing is evident; but, if they intersect in ${\displaystyle V}$, draw ${\displaystyle ECF}$ parallel to ${\displaystyle VD}$, the line most remote from ${\displaystyle C}$; then, by the first part, ${\displaystyle EC=CF}$, and ${\displaystyle AD:AC::VD:EC}$ or ${\displaystyle CF}$. Therefore ${\displaystyle AD:AC::BD:CB}$. q. e. d.

   • 1999, William Dunham, “Euler and Analytic Number Theory”, in Euler: The Master of Us All (The Dolciani Mathematical Expositions; 22), [Washington, D.C.]: Mathematical Association of America, →ISBN, page 64:

     By Cases 1 and 2, we see that any finite collection of ${\displaystyle 4k-1}$ primes cannot contain all such primes. Thus there are infinitely many primes of this type. ¶ Q.E.D.

2. (by extension) Used to indicate that an argument or proposition is proved by the existence of some fact or scenario.
   • 1809, Diedrich Knickerbocker [pseudonym; Washington Irving], chapter IV, in A History of New York, from the Beginning of the World to the End of the Dutch Dynasty. […], volume I, New York, N.Y.: Inskeep & Bradford, […], OCLC 426050984, book I, page 42:

     That this part of the world has actually been peopled (Q. E. D.) to support which, we have living proofs in the numerous tribes of Indians that inhabit it.

   • 1870, Benedict de Spinoza [i.e., Baruch Spinoza]; R[obert] Willis, “[The Ethics.] Part I.—Of God.”, in Benedict de Spinoza; His Life, Correspondence, and Ethics, London: Trübner & Co., […], OCLC 1021877894, page 418:

     PROP[OSITION] VI. One substance cannot be produced by another substance. ¶ Demonst[ration]. In the preceding proposition we have seen that there cannot in the nature of things be two Substances of the same attribute, or that they have anything in common (by Prop. II.); and so (by Prop. III.) one cannot be the cause of, or be produced by, another: q. e. d.

   • 1908 July, C. H. Claudy, “Tanks”, in Frank V. Chambers, editor, The Camera: An Illustrated Magazine Devoted to the Advancement of Photography, volume 12, number 7, Philadelphia, Pa.: The Camera Publishing Company, page 252:

     Of all those on the market, Eastman's is the only one with a solution tight, locking cover. ¶ Q. E. D.

使用する際の注意点

When used to end a mathematical proof, QED is somewhat dated or traditional; modern textbooks often use the graphical symbol ∎ (the halmos または tombstone) instead. Other languages generally use a vernacular abbreviation, such as French CQFD (“ce qu'il fallait démontrer”) or Portuguese C.Q.D. (“como queríamos demonstrar”).

別の表記

• QED, Q. E. D., q. e. d., q.e.d.

関連する語

• QEF

名詞

Q.E.D. (複数形 Q.E.D.s)

1. Some fact or scenario that proves an argument or proposition; a justification.
   • 1867, Ouida [pseudonym; Maria Louise Ramé], “The General’s Match-making; or, Coaches and Cousinship”, in Cecil Castlemaine’s Gage, and Other Novelettes, London: Chapman and Hall, […], OCLC 35980974, page 233:

     We all go in for the dolce here except you, and you're such a patent machine for turning out Q.E.D.s by the dozen, that you can no more help working than the bedmaker can help taking my tea and saying the cat did it, […]

   • 1936 April, Nina Melville, “The Theatre”, in V[ictor] F[rancis] Calverton [pseudonym; George Goetz], editor, The Modern Monthly: An Independent Journal of Radical Opinion, volume IX, number 9, OCLC 4756578, page 27, column 1:

     The same thing, I believe, that was lacking in Let Freedom Ring—lack of dramatic integration, lack of dramatic intensity, lack of all those elements, call them what you will, which make a play a play and not a tract, and certainly not a Q.E.D. thesis.

別の表記

• QED

参照