Exploring $\equiv$, =>, and <=> Symbols

[Dethrone]

First day of school and I don't have much work. So I got bored and read ahead a bit. From a textbook my prof wrote:

Can someone explain why they used the "$\equiv$"? I think it means "equivalent", but I'm not sure, but when are times you want to use that symbol rather than "equals"? What's the difference between the two, and any examples of other usages of the symbol, or when you would use one over the other?

Other symbols of confusion: "=>". I generally use this to mean "implies", like $3x+y=6 => y=6-3x$. How about that versus "<=>" which I also don't see too often. Any other symbols of importance? (Wondering)

 

[evinda]

First day of school and I don't have much work. So I got bored and read ahead a bit. From a textbook my prof wrote:Can someone explain why they used the "$\equiv$"? I think it means "equivalent", but I'm not sure, but when are times you want to use that symbol rather than "equals"? What's the difference between the two, and any examples of other usages of the symbol, or when you would use one over the other?

Other symbols of confusion: "=>". I generally use this to mean "implies", like $3x+y=6 => y=6-3x$. How about that versus "<=>" which I also don't see too often. Any other symbols of importance? (Wondering)

Two expressions are equal for some specific value of the
variable, when they yield the same number,but they are equivalent when
they are equal for any value of the variable. Two equations are equivalent when they have the same solution set.It is not used to say that two equations are equal.
Two sets are equivalent when they have the same number of
elements,but they are equal if they have exactly the same elements.

$\Leftrightarrow$ is the symbol of "if and only if",at which the truth of either one of the connected statements requires the truth of the other, either both statements are true, or both are false

 

[Euge]

In the context of the above picture, the $\equiv$ symbol means "by definition".