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This question was given to us in my discrete mathematics class and will show up on the exam, i'm having trouble understanding it and was hoping someone here could clarify.

What I understand thus far:

A truth function receives a series of values, True and False values, and returns a True or False value. In this case for part (a) of the question, we are constructing an algorithm that will return a statement S whose truth function is similar to our original truth fucntion, f.

part (b) wants us to prove this statement returns a combination of values that is identical to the truth function we use and their output value is identical.

This question is giving us no values to use, i'm assuming the question wants us to construct our own algorithm for finding a statement and proving one of it's statements in the function is always true and identical to the original function. Could anyone help point me in the correct direction for constructing one?

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    How is that function defined (domain, image sets) and fed to the algorithm? What is the format of the statement that has to be returned by the algorithm? A string, a syntax tree? – 2017-02-02
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    The values (which are non-existent) are being fed to the truth function as a boolean function, the domain is the set of all the ordered pairs of T's and F's. The statement should look something like S = T ^ F ^ T ^ F = T/F. Sorry if i'm not being clear enough to understand... – 2017-02-02
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    In your title who's: $\rightarrow$ whose (btw, are you in the "Who's Who" ?) – 2017-02-02
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    Ah, thanks lol. Guess I learned something else valuable in this post. – 2017-02-02
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    @Nelsoned There are many ways to represent a function, like a piece of JavaScript, a table of (argument, function values). What do you use? – 2017-02-02
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    A table of rows and columns representing a series of true and false values, kind of like a truth table – 2017-02-02

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Note that every boolean function $f: \{\top,\bot\}^n \to \{\top,\bot\}$ can be written as $$f(p_1,\ldots,p_n)=\bigvee_{(q_1,\ldots,q_n) \in A} \bigwedge_{i=1}^n g(q_i,p_i)$$ where $$A=\{ (q_1,\ldots,q_n) \in \{\top,\bot\}^n \mid f(q_1,\ldots,q_n) = \top \},$$ $$g(q,p)=\begin{cases} p, & \text{if } q=\top \\ \neg p, & \text{if } q=\bot \end{cases}.$$

Indeed, inner conjuction $$\bigwedge_{i=1}^n g(q_i,p_i) = \begin{cases} \top, & \text{when } (p_1,\ldots,p_n)=(q_1,\ldots,q_n) \\ \bot, & \text{otherwise} \end{cases}$$ so outer disjunction is true exactly for those $(p_1,\ldots,p_n)$ that belong to $A$, and false otherwise.

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    This is quite similar to what my professor did during his lecture, could you verbalize what's happening in this algorithm? – 2017-02-02
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    @Nelsoned Can you now see what is going on? – 2017-02-02
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    Yes! I see it now, thank you! – 2017-02-02