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So I'm reading "A Mathematical introduction to logic" by Herbert.E.enderton.

In the chapter on sentential logic, it used a phrase "Boolean function realized by a wff, $\alpha$"

So I'm abit confused, what exactly do we mean by "wff $\alpha$ realizes a Boolean function"

If a concrete example can be given, that would be so much help.

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A Boolean function is any function that takes in a number of truth-values, and returns a truth-value. So, for example, suppose we have a function that takes the truth-values of A, B, and C, and that returns True whenever at least two of these are True (and otherwise returns False) Then we can use the following wff to 'realize' that function:

$(A \land B) \lor (A \land C) \lor (B \land C)$

Of course, there are other wff's that realize that same function. They would all be considered equivalent.

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    So what you are saying is that "realization" in this sense is analogous to construction of a function say $F : \mathbb{R} \rightarrow \ \mathbb{R}$ where $F(x) = x^2$, where we want a function to take a real value $x$ and map it to $x^2$. Here using your example we want a Boolean function to return true when majority of the 3 variables A, B, C is true. Thus we construct Such a Boolean function as per your construction. – 2017-02-02
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    @some1fromhell Exactly. And sometimes the function we want is easy, so we have a simply formula (wff) that captures that function, but other times it is a more complicated function, and we'll need a more complex formula. But interestingly, it turns out that no matter how complicated the function is, we can always capture it using some wff that uses just $\land$, $\lor$, and $\neg$! – 2017-02-02
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    Thank you, you clear my doubts. – 2017-02-02