Mathematics Educators Asked on September 6, 2021

In schools, many students learn the usage of "$therefore$" and "$because$" in proofs. Such three-dot notation are popular in many high-school books and exams, but are almost never used in university-level texts. (It seems that, at degree level, this notation only appears in some books about mathematical logic.)

Very often, it is somewhat awkward to use "$therefore$" and "$because$" for proofs, because modus ponens, the most commonly used principle of deduction, contains **three** parts, while "$therefore$" and "$because$" are just **two** symbols. Modus ponens states that from $ARightarrow B$ and $A$ we could deduce $B$, so the three parts are: $ARightarrow B$, $A$ and $B$.

We will of course write $B$ after "$therefore$", but it is a good question where to put $ARightarrow B$ and $A$. We may either put both $A$ and $ARightarrow B$ after "$because$", or put $A$ after "$because$" and $ARightarrow B$ in brackets after "$therefore B$".

In the end, the three-dot notation does not make the logic structure entirely clear. "$therefore $" clearly indicates the conclusion, but the meaning of "$because$" is not entirely clear – it could be either a theorem $ARightarrow B$ or a condition $A$. Sometimes, $A$ is too **long** (takes too many words) to be written out fully, which causes confusion.

**Is there any better alternative to the three-dot notation?** It is, after all, completely clear to just write everything in words.

The context isn't entirely clear so I'll assume this is about teaching. Then, I support Pedro's answer but also want to add that doing both verbal and symbolic versions may be a good idea. For example:

Theorem. A polynomial has a higher order than another if and only if its degree is higher.

In other words, for any two polynomials $P$ and $Q$, we have: $$P=o(Q) Longleftrightarrow deg P<deg Q, .$$

Answered by Peter Saveliev on September 6, 2021

*Is there any better alternative to the three-dot notation?*

The usual general advice is to use words instead of symbols.

The best notation is no notation; whenever it is possible to avoid the use of a complicated alphabetic apparatus, avoid it. A good attitude to the preparation of written mathematical exposition is to pretend that it is spoken. Pretend that you are explaining the subject to a friend on a long walk in the woods, with no paper available; fall back on symbolism only when it is really necessary.

(Paul Halmos,

How to Write Mathematics, p. 40.)

This applies particularly to the three-dot notation.

Do not misuse the implication operator ⇒ or the symbol ∴. The former is employed only in symbolic sentences; the latter is not used in higher mathematics.

Bad: a is an integer ⇒ a is a rational number.

Good: If a is an integer, then a is a rational number.

Bad: ⇒ x = 3.

Bad: ∴ x = 3.

Good: hence x = 3.

Good: and therefore x = 3.Bad Theorem. n odd ⇒ 8|n² − 1.

Bad proof.

n odd ⇒ ∃j ∈ Z, n = 2j + 1;

∴ n² − 1 = 4j(j + 1);

∀j ∈ Z, 2 | j(j + 1) ⇒ 8 | n² − 1This is a clumsy attempt to achieve conciseness via an entirely symbolic exposition.Combining words and symbols and adding some short explanations will improve readability and style.

Answered by Pedro on September 6, 2021

0 Asked on January 6, 2022 by opal-e

2 Asked on December 28, 2021

20 Asked on November 22, 2021 by markus-klein

mathematics in daily life secondary education statistics undergraduate education

2 Asked on November 17, 2021

11 Asked on September 6, 2021

5 Asked on September 6, 2021

4 Asked on September 6, 2021 by zuriel

calculus student motivation undergraduate education undergraduate research vectors

2 Asked on September 6, 2021

11 Asked on September 6, 2021 by user13395

4 Asked on September 6, 2021 by eleven-eleven

1 Asked on September 6, 2021

8 Asked on September 6, 2021 by welshgandalf

4 Asked on September 6, 2021 by timothy

8 Asked on September 6, 2021 by madscientist

18 Asked on September 6, 2021 by axel-tong

6 Asked on September 6, 2021 by a-goodier

2 Asked on September 6, 2021 by nothing

2 Asked on September 6, 2021

2 Asked on September 6, 2021 by ari

education research english language learners mathematical pedagogy reference request

Get help from others!

Recent Answers

- Peter Machado on Why fry rice before boiling?
- Jon Church on Why fry rice before boiling?
- haakon.io on Why fry rice before boiling?
- Lex on Does Google Analytics track 404 page responses as valid page views?
- Joshua Engel on Why fry rice before boiling?

© 2022 AnswerBun.com. All rights reserved. Sites we Love: PCI Database, MenuIva, UKBizDB, Menu Kuliner, Sharing RPP, SolveDir