Qed Definition

Decoding "QED": Understanding the Significance and Applications of "Quod Erat Demonstrandum"

In the world of mathematics, logic, and formal argumentation, the abbreviation "QED" holds a prominent position. Standing for "Quod Erat Demonstrandum," a Latin phrase meaning "that which was to be demonstrated," QED signifies the conclusion of a mathematical proof or logical argument. Understanding its meaning and proper usage transcends mere academic pedantry; it speaks to the core principles of rigorous reasoning and clear communication. This article will delve into the definition of QED, address common misconceptions, and explore its practical applications across various disciplines.

1. Understanding the Core Meaning of QED

At its most basic level, QED marks the end of a logical progression where a hypothesis or theorem has been successfully proven. It signals that the initial statement has been definitively established through a chain of logically sound deductions. This doesn't merely mean that the conclusion seems plausible; it means that its truth is demonstrably linked to established axioms or previously proven theorems. The elegance of QED lies in its concise yet powerful declaration of a successful proof.

For instance, consider the simple Pythagorean theorem: a² + b² = c² (for a right-angled triangle). A QED at the end of a geometric proof would indicate that the theorem has been rigorously proven using only axioms of geometry and established logical rules.

2. QED vs. Other Conclusive Statements

While QED is often associated with mathematical proofs, its broader significance extends to any rigorous argumentative process. However, it's crucial to distinguish QED from other concluding phrases that might seem similar. For example:

"Therefore" or "Thus": These words indicate a logical consequence but don't necessarily denote a complete and rigorous proof. They might signify a conclusion drawn from limited evidence or incomplete reasoning.
"Hence" or "Consequently": Similar to "Therefore," these terms suggest a conclusion but lack the formal weight of a fully demonstrated proof.
"It follows that": This phrase indicates a deductive step, but it doesn't necessarily signal the end of the entire argumentative process.

QED, on the other hand, specifically implies the successful completion of a rigorous demonstration, leaving no room for doubt or further justification.

3. The Practical Application of QED Beyond Mathematics

While deeply rooted in mathematics, the principle underlying QED—the rigorous demonstration of a statement—finds relevance across various fields:

Law: Legal arguments often strive for a QED-like conclusion, aiming to prove a case beyond reasonable doubt. The presentation of evidence and logical reasoning seeks to establish the truth of a claim.
Computer Science: Algorithm verification requires rigorous proof that the algorithm functions as intended under all possible inputs. QED-like verification ensures the correctness and reliability of software.
Philosophy: Philosophical arguments often aspire to reach conclusions backed by logical reasoning and evidence. While absolute certainty might be elusive, the pursuit of a QED-like conclusion guides the process.
Scientific Research: Scientific papers often employ QED-like conclusions following experimental evidence and statistical analysis, demonstrating the validity of a hypothesis.

4. Misconceptions and Challenges in Using QED

Despite its seeming simplicity, using QED correctly requires careful consideration:

Overuse: Using QED inappropriately can diminish its impact. It should only be used when a rigorous proof has genuinely been established.
Ambiguity: In informal settings, the meaning might be less precise. To avoid ambiguity, always ensure that the context clarifies the scope and rigor of the "proof."
Incomplete Proofs: A common mistake is declaring QED prematurely, before all steps in the argument have been fully justified.

5. Step-by-Step Example: A Simple Proof

Let's illustrate with a simple algebraic proof:

Theorem: If x + 2 = 5, then x = 3.

Proof:

1. Given: x + 2 = 5
2. Subtract 2 from both sides: x + 2 - 2 = 5 - 2
3. Simplify: x = 3
4. QED

This concise proof demonstrates the application of QED to conclude a simple argument. Each step is logically sound and leads directly to the desired conclusion.

Summary

QED, signifying "Quod Erat Demonstrandum," is more than just a Latin abbreviation; it represents the culmination of rigorous logical reasoning. Its use extends beyond mathematics, embodying the pursuit of demonstrable truth across disciplines. While seemingly simple, its proper application requires careful consideration of context, rigor, and the completeness of the argument. Understanding and utilizing QED fosters clarity, precision, and a deeper appreciation for the power of logical argumentation.

FAQs

1. Can QED be used in informal settings? While technically it refers to formal proof, it's sometimes used informally to playfully signal the conclusion of an argument, though this usage is less precise.

2. What is the difference between QED and a conjecture? QED signifies a proven statement, while a conjecture is a statement that is believed to be true but lacks a formal proof.

3. Is there a modern equivalent to QED? Many authors use symbols like a square (∎) or a triangle (△) instead of QED to mark the end of a proof, representing the same function.

4. Can a proof be considered complete without a QED? While not strictly necessary in all cases, the use of QED or a similar symbol is a clear and universally understood indication of the end of a formal proof. Omitting it can lead to ambiguity.

5. What happens if a proof using QED is later found to be flawed? The QED would then be considered invalid, and the "proof" would require revision or retraction. The flaw might lie in a single step, an incorrect assumption, or a fundamental gap in reasoning.

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Minimal Modal Logic and the Class of all Kripke Models From these two, we get $\vdash\neg\neg\phi$ and then $\vdash\phi$ which contradicts the premis. $\qquad$ QED. Definition 1. A maximally consistent set is a consistent set that every proper superset of it is inconsistent (equivalently, for every formula $\phi$, either $\phi$ is in the set or $\neg\phi$). Definition 2.

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Why are projective morphisms closed? - Mathematics Stack … qed We consider the set $\Omega_{\infty} = \Omega \cup \{\infty\}$, where $\infty$ is an element which does not belong to $\Omega$. Let $\rho\colon \Omega_{\infty} \rightarrow \Omega_{\infty}$ be the map defined as follows.

probability theory - Understanding a proof of the strong Markov ... 27 May 2015 · (b) The "only if" direction follows from the definition of weak convergence. The "if" direction follows from the "only if" direction together with part (a). QED. Theorem: Uniqueness of measures [Theorem 1.5, p. 8] Let $\mu$ and $\nu$ be two $\sigma$-finite measures on the $\sigma$-algebra $\mathcal{A}$. If $\mu$ and $\nu$ coincide on a $\pi ...

Why does drawing $\\square$ mean the end of a proof? 21 Apr 2017 · I had always used QED (quod erat demonstrandum) at the end of a proof until I was introduced to using Halmos' symbol. Since it was a 4-sided symbol, my mind has always associated this "quad" symbol with "quod".

Dedekind's theorem on an integrally closed algebra over a … Stack Exchange Network. Stack Exchange network consists of 183 Q&A communities including Stack Overflow, the largest, most trusted online community for developers to learn, share their knowledge, and build their careers.

Basic question about mod - Mathematics Stack Exchange The Wikipedia article on the modulo operation is a good source for what symbol and which definition is used in each of a variety of languages. $\endgroup$ – Isaac Commented Jun 28, 2011 at 5:12

What are the advantages of ending a proof with “QED”? 31 May 2019 · As you know “QED” means “quod erat demonstrandum”, that is “what was to be shown”. It makes proofs more readable. Modern textbooks often conclude proofs with “which concludes the proof” or “which proves the assertion” or something. It’s the same thing, “QED” and the square are just convenient shorthands. $\endgroup$ –

terminology - What is the meaning of the expression Q.E.D.? Is it ... 19 Feb 2015 · From A Comprehensive Dictionary of Mathematics by Roger Thompson: "quod erat demonstrandum" (Latin) -- This stems from medieval translators' habitual tendency of translating the Greek for "this was to be demonstrated" to the Latin phrase above.

general topology - How can I prove formally that the projective … 12 Dec 2014 · I think in this case, and in may others, it is way easier to use the following equaivalent definition of Hasudorff: Definition: A topological space X is Hausdorff iff the Diagonal $\,\Delta_X:=\{(x,x)\in X^2\}\,$ is closed in the product space $\,X\times X\,$