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Look at below formulas:

$(\forall a)(a

Now if one considers $[(\forall a)(a

Can we claim anything about overall occurrence of $a$ and $b$ to be bound or free?

In other words, is $[(\forall a)(a

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    It is not a *sentence* because $b$ is free in the left disjunct and $a$ is free in the right one. – 2017-01-15
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    Although your $[(\forall a)(a – 2017-01-15
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    As you correctly write, it is an *occurrence* of a variable that is free or bound, and not the variable. – 2017-01-15
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    @MauroALLEGRANZA: Not for some authors; see my answer. =) – 2017-01-17
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    @user21820 - agreed... but we can match the two "points of view" : a variable $x$ is *free* in a formula $\varphi$ if it has one or more free occurrences; otherwise, it is *bound*. – 2017-01-17
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    @MauroALLEGRANZA: Yup certainly, though I would refrain from using the word "bound" since a variable might be not free might simply because it does not occur in the formula at all. – 2017-01-17

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Some textbooks (such as Rautenberg's linked from this post) do define the set of free variables of a formula, such that it include variables that occur at least once in the formula in an unbound state. In particular $free(\ \forall a(a