Equal Sign With Three Lines

I think you know the i I mean. I don't sympathise the point in it though; it is used to say 'equal to at all values', but surely a '=' sign does exactly the same thing, as long equally at that place aren't whatever restrictions (equally in when x < 5)?

No. Say someone has written 10 + 2 = 2x - 2. Is this true for all values of x? Information technology's not true for whatsoever value of x other than 4. In contrast, if ten^2 + 2x + 1 = (10 + 1)^2 is written, and so this is true for all values of x. Then it should be written $x^2 + 2x + 1 \equiv (x + 1)^2$ . It's called an identity.

Edit: Strictly, $1 + 1 \equiv 2$ . But mathematicians are lazy and leave off the extra nuance.

(Original post by Glutamic Acid)
No. Say someone has written x + two = 2x - two. Is this true for all values of x? It'due south non true for whatsoever value of ten other than 4. In contrast, if x^2 + 2x + one = (ten + 1)^2 is written, then this is true for all values of 10. So it should be written $x^2 + 2x + 1 \equiv (x + 1)^2$ . It's called an identity.

:ditto:

This is a skillful question. I believe the note is a shorthand used because sometimes we demand to arrive explicit that variables require quantification. In A-level maths, the quantification is usually left implicit. Nevertheless, in some cases, such as those equals signs with three lines, it is made clearer. The divergence is betwixt $\exists x (P(x)) = Q(x))$ and $\forall x (P(x) = Q(x))$ . (Where $\exists$ stands for "there exists", and $\forall$ stands for "for all".) We oftentimes utilise the context to inform the reader which example nosotros are considering in our algebraic manipulation but, sometimes, if we want to make it clearer, we write $P(x) \equiv Q(x)$ to denote $\forall x (P(x) = Q(x))$ . is used (confusingly) to denote both $\exists x (P(x) = Q(x))$ and $\forall x (P(x) = Q(x))$ , although some may want to simply utilise = for the existential quantifier, to make their work clearer.

In the instance of ane+ane=2, it is true that both $\exists x (1+1=2)$ and $\forall x (1+1=2)$

(Original post past Kolya)
This is a expert question. I believe the notation is a shorthand used because sometimes we demand to make information technology explicit that variables require quantification. In A-level maths, the quantification is ordinarily left implicit. All the same, in some cases, such every bit those equals signs with three lines, it is fabricated clearer. The deviation is between $\exists x (P(x)) = Q(x))$ and $\forall x (P(x) = Q(x))$ . (Where $\exists$ stands for "at that place exists", and $\forall$ stands for "for all".) We often utilise the context to inform the reader which case we are because in our algebraic manipulation but, sometimes, if we want to make it clearer, we write $P(x) \equiv Q(x)$ to denote $\forall x (P(x) = Q(x))$ . is used (confusingly) to announce both $\exists x (P(x) = Q(x))$ and $\forall x (P(x) = Q(x))$ , although some may want to only employ = for the existential quantifier, to make their work clearer.

In the case of 1+i=2, information technology is true that both $\exists x (1+1=2)$ and $\forall x (1+1=2)$

Nice explanation. You have one as well many brackets for ane of your earlier expressions though.

These days when I meet $\equiv$ , I retrieve 'modulo', as I suppose quite a lot of others do on here likewise.

It could also be used every bit "definition", for example, ascertain $x \equiv y + 1$ .

(Original post past im so bookish)
I still don't get it

Ask yourself 'Does this work for whatsoever value of x?' If it does, then theoretically, you should use the identity sign (although often, it won't be used). If it doesn't you can merely use the equals sign.

For case:

2x + one = 3x

When 10 = 1, this works. Simply when ten = two, it doesn't ( $4 + 1\not=6$ )

However, using the earlier example;

This works for any value of x. Go on, endeavor information technology... I cartel you

Therefore, information technology could, and indeed should exist written as $(x + 1)^2\equiv x^2 + 2x + 1$

(Original post past nigel_s)
Ask yourself 'Does this work for whatsoever value of x?' If information technology does, and so theoretically, you should use the identity sign (although often, it won't exist used). If it doesn't you lot tin only apply the equals sign.

For example:

2x + 1 = 3x

When x = 1, this works. But when 10 = two, it doesn't ( $4 + 1\not=6$ )

However, using the earlier example;

This works for any value of x. Become on, try it... I dare y'all

Therefore, it could, and indeed should be written equally $(x + 1)^2\equiv x^2 + 2x + 1$

so in the 1st example x can only be 2, but not four, 5.vi, 78 = and then then you lot would use the two line sign,

merely in your 2d example, for x, any number can be put in and it'll still be right.

am I right?

cheers

Yeah that'south right - at least that's how I understand it.

Another example would be:
$(x+1)(x-1)\equiv x^2 - 1$

The two line equals sign is a pair of scales. What is on each side may be different just has to have the aforementioned weight/value. The objective is to find the variable.

The three line equal sign, the identity sign, is like a mirror. What is on each side is identically the same, just represented in a dissimilar way.