Difference Between Double Implications and Equal Signs

Question

I always thought equal signs (=), indicated double implications, however after thinking about this example, I am very confused.

For example, the Theorem (in Linear Algebra):

L($\vec{0}$) = $\vec{0}$, where L is a linear mapping can be described as:

In the equation L($\vec{x}$) = $\vec{y}$, if $\vec{x}$ = $\vec{0}$, then $\vec{y}$ = $\vec{0}$.

However, the other implication is not true, which is if $\vec{y}$ = $\vec{0}$, then $\vec{x}$ = $\vec{0}$.

Do equals not always indicate double implications?

Thanks

Answer 1

In logic, the equal sign expresses equality between terms, i.e. "names" for objects, like e.g. $2=1+1$.

Bi-conditional (i.e. double implication) is a connective "connecting" formulas, like e.g. $p \leftrightarrow q$.

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Regarding the "equation" : $L(0)=0$, maybe it is only the expression of a function $L$ that for input (argument) $0$ returns output (value) $0$.

In this case, it is an equality as expressed above : we have a "name" for an object (a number), i.e. the value of the function $L$ for argument $0$, and the exprssion asserts that that value is (equal to the number) $0$.

Answer 2

"$L(x)=y$" means:

Term $L(x)$ equals term $y$.

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"If $A$ then $B$" means:

Proposition $A$ implies proposition $B$ or

Whenever proposition $A$ is true then Proposition $B$ must also be true

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For the latter it does not matter, if $A$ or $B$ include some terms containing equality signs or not.