Is this true for biconditional statement? Given
A = At least ten people are there
L = Lecture will be given
Does A $\iff L$ means that lecture will be given only when at least ten people are there, means there cannot be other reason for lecture to be given and nothing else. And if ten people are not there then lecture will not be given at any cost.
Is this true ?
Material biconditional (symbol $X \leftrightarrow Y$) contains no idea of possibility or impossibility. It only means that, as a matter of fact, propositions $X$ and $Y$ do not have different truth values, that is, that they are either both true or both false. So as soon as (for a reason or another) the lecture is given and (for a reason or another) there are at least $10$ people, it is true that  $L \leftrightarrow A$.
Note: both material implication and bi-implication are very poor statements as to their informational contents

Logical equivalence (symbol: $\iff$, or $\equiv$) is more demanding. It requires not only that $X$ and $Y$ have the same truth value, but also that $X$ and $Y$ cannot possibly have different truth values. Or, if you prefer, it requires that $X$ and $Y$ have necessarily the same truth value.
So $X \iff Y$ standardly means that the material biconditional $X \leftrightarrow Y$ is always true, in all (logically) possible cases.

Normally, the kind of possibility / necessity invoked here is a logical one. But, in the example you take, it seems the necessity is more of a &quot;legal&quot; sort. The case you are considering is: &quot;due to the rules of the university,
(a) the fact that there are at lest 10 students is necessarily a sufficient condition for the lecture to be given
and
(b) the fact that there are at least 10 students is necessarily a necessary condition for the lecture to be given&quot;.


Reference: Seymour Lipschutz, Schaum's Outline Of Set Theory (the chapter on the &quot;Algebra Of Propositions&quot;), at archive.org.
Material biconditional (symbol $X \leftrightarrow Y$) contains no idea of possibility or impossibility. It only means that, as a matter of fact, propositions $X$ and $Y$ do not have different truth values, that is, that they are either both true or both false. So as soon as (for a reason or another) the lecture is given and (for a reason or another) there are at least $10$ people, it is true that  $L \leftrightarrow A$.
Note: both material implication and bi-implication are very poor statements as to their informational contents

Logical equivalence (symbol: $\iff$, or $\equiv$) is more demanding. It requires not only that $X$ and $Y$ have the same truth value, but also that $X$ and $Y$ cannot possibly have different truth values. Or, if you prefer, it requires that $X$ and $Y$ have necessarily the same truth value.
So $X \iff Y$ standardly means that the material biconditional $X \leftrightarrow Y$ is always true, in all (logically) possible cases.

Normally, the kind of possibility / necessity invoked here is a logical one. But, in the example you take, it seems the necessity is more of a &quot;legal&quot; sort. The case you are considering is: &quot;due to the rules of the university,
(a) the fact that there are at lest 10 students is necessarily a sufficient condition for the lecture to be given
and
(b) the fact that there are at least 10 students is necessarily a necessary condition for the lecture to be given&quot;.


Reference: Seymour Lipschutz, Schaum's Outline Of Set Theory (the chapter on the &quot;Algebra Of Propositions&quot;), at archive.org.