What would be the symbolic representation of both statements?
Those definitions of the validity of a semantic argument are equivalent.
A conclusion is a logical consequence of the argument's premises exactly when it is impossible for that conclusion to be false in any interpretation where all of those premises are true.
Let $\Gamma$ be a set of statements and $\varphi$ a single statement. Then we indicate that an argument with premises of $\Gamma$ and conclusion of $\varphi$ is semantically valid by writing:
$$\Gamma\vDash\varphi$$
How can I spot that premises and conclusion are connected, i.e. the conclusion follows from the premises? Are there a specific set of accepted form regarded as valid?
You should have some language and theory, or semantics, by which you can interpret and evaluate the truth of statements.
Take $p$ and $q$ as literals in the language, which represent propositions with values of true or false, and $\to$ as a logical connective in the language, such that the statement $p\to q$ is evaluated as false only when $p$ is true but $q$ false. The argument that concludes $q$ under premises of $p$ and $p\to q$ , is semantically valid. The reason being, that it is impossible to value $q$ as false when valuing $p$ and $p\to q$ as true.
$$p, p\to q\vDash q$$
