What does the sign "$=$" exact meanings?

Question

How can I understand the sign "$=$" from the following expression:

$$\mathcal{o}f((x))=\mathcal{o}f((x))+\mathcal{o}f((x));$$ $$\mathcal{o}(kf((x)))=\mathcal{o}(f(x));$$ $$t\mathcal{o}(f((x)))=\mathcal{o}(f(x)).$$ ( $k$ and $t$ are constants,$\mathcal{o}$ is little $o$ Notation )

Someone say the sign "$=$ " only means "$\subseteq$ " ; others confirm" the sign "$=$"has the same meaning as the equivalence of two sets (i.e.Let two sets $A,B$:$A=B\Leftrightarrow A$$\subseteq $$B$ and $A$$\supseteq $$B$ )."But what does the sign "$=$" exact meaning?

possible duplicate of What are the rules for equals signs with big-O and little-o? — JMoravitz, Apr 05 '15 at 10:55
@JMoravitz:According to your useful link,the sign "$=$" only means "$\subseteq$". — , Apr 05 '15 at 11:10
If the abuse bothers you and you want to try to help put an end to it, another common notation I have seen is $\asymp$ written as \asymp. — JMoravitz, Apr 05 '15 at 11:29
@JMoravitz To mean...what? I have only seen that notation used for logarithmic equivalence. — Ian, Apr 05 '15 at 12:01
Related/duplicate question: http://math.stackexchange.com/q/514263/11994. Personally I think my answer there is the simplest explanation I've seen for this notation, and a longer more precise version is in http://math.stackexchange.com/a/86088/11994. — MarnixKlooster ReinstateMonica, Apr 05 '15 at 15:33

score 3 · Answer 1 · answered Apr 05 '15 at 11:26

3

It should be read as a $\subseteq$, inclusion only in the one direction.

Taken from Concrete Mathematics by R. Graham

enter image description here

answered Apr 05 '15 at 11:26

JMoravitz

80,908

1

I have never liked this abuse of notation, and prefer to write e.g. $f(n) \in \mathcal O(n^2)$. – Math1000 Apr 05 '15 at 11:46

What does the sign "$=$" exact meanings?

1 Answers1