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Anar22 · Jan 7, 2012, 07:23 AM

I want to know this symbol math
It is similar to the symbol is equal to
But contains a third line above

corrigan · Jan 7, 2012, 05:46 PM

does it look like this -----> $\equiv$ . It's a general equivalence relation symbol, that is it's used to indicate an equivalence between the two objects on either side of the relation symbol. The conditions on an equivalence relation are 1) reflectivity, 2) symmetry, and 3) transitivity, so you would need to know the definition of the equivalence in question.

In logic it's used as an equals sign in the metalanguage. That is it's used to mean equals in the everyday language we talk in, and not to be confused with the mathematical structure being discussed. I've also seen it used the other way around. Where did you see it and what context was it in?

TUT317 · Jan 7, 2012, 06:51 PM

Originally Posted by corrigan

does it look like this -----> $\equiv$ . It's a general equivalence relation symbol, that is it's used to indicate an equivalence between the two objects on either side of the relation symbol. The conditions on an equivalence relation are 1) reflectivity, 2) symmetry, and 3) transitivity, so you would need to know the definition of the equivalence in question.

In logic it's used as an equals sign in the metalanguage. That is it's used to mean equals in the everyday language we talk in, and not to be confused with the mathematical structure being discussed. I've also seen it used the other way around. Where did you see it and what context was it in?

Hi corrigan,

Very good explanation. I know Anar22 was asking about mathematics but an example in language might be.

Smith is a bachelor necessarily implies that he is a male. It also necessarily implies he is unmarried.

Smith is a bachelor => unmarried male.

This is off topic and is not related to what you were talking about but it gives me a change to have a whinge about metalanguage in modern educational theory. 'Metalanguage' and 'metacognition' in modern educational theory shows a complete misunderstanding and a complete lack of understanding of the terms. Anyway, I better not go there.

Tut

Anar22 · Jan 8, 2012, 08:35 AM

hi every one

What real difference between this code and =?
And when we use both؟

corrigan · Jan 9, 2012, 03:11 PM

While I was typing this I also thought of seeing $\equiv$ is used in modular arithmetic. For example, $9\equiv 15 (\ mod{6})$ . Now that I think about it, that is probably the most popular use for it. Modular arithmetic is used when counting by subsets (usually numbers). So $9\equiv 15 \ mod{6}$ means that 9 and 15 are the same when we count by multiples of 6. to put it another way $9 = 3 + 6 \cdot k \equiv 3 + 6\cdot k^{\prime} =15$ , where $k$ and $k^{\prime}$ are integers. So 9 and 15 are both "equivalent" to 3 plus some multiple of 6, but they are not "equal".

I'm not sure what you mean by "real difference". There are many (actually an infinite number) of equivalence relations, equals is simply one of them. The real difference would depend on the definition involved. All something needs to be an equivalence relation is that it satisfies the three conditions I gave you. Let me give you a non-arithmetic example:

Let $X$ and $Y$ be words, and let $X \equiv Y$ if and only if they have the same number of vowels. So $green \equiv blue$ since they both have two vowels, and $red \not\equiv white$ since one has two vowels and the other has one vowel. I think the best way to demonstrate the three properties is by example, so
for reflectivity, $green \equiv green$ .
for symmetry, $green \equiv blue$ implies $blue \equiv green$ .
for transitivity, since $green \equiv blue$ and $blue \equiv white$ , then $green \equiv white$ .
So it fits the definition of equivalence relation. I could give you a more rigorous example, but I don't know your level of math.

I could be more helpful if I knew what context you saw $\equiv$ . If it was modular arithmetic, I know how much trouble students can have with that. I'll be watching this thread.