You can approach that by "simple" stepwise elimination:

1: (a AND b) OR (a AND c) OR (b)

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We know that the term evaluates true iff one of the OR terms evaluates to a true value. So if we start looking at b, we know that the term evaluates true whenever b has a true value. Therefore, we can rewrite it as

2: b OR (X)

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where X is the result of the evaluation of the term 1: with b a false value:

3: b OR (a AND false) OR (a AND c) OR (false)

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Now, we can apply one round of simplification again, replacing Y AND false by false and false OR Z by Z (modulo commutativity):

4: b OR (a AND c)

How one would really encapsulate the reduction of such terms to a "appealing" form in the sense that it has the fewest number of variables or whatever, I don't know, but I guess the idea is to split up any term X OR Yinto the form (true OR Y) OR (false OR Y)

and then simplifying again, for Y being an atomar truth variable. Whenever you can't find a suitable Y, you have to create a synthetic truth variable Y' := Y'' AND Y''', where Y'' and Y''' are (possibly synthetic) truth variables. This amounts more or less to building the complete truth table.

In reply to Re^3: simplify logical equations by Corion
in thread simplify logical equations by thealienz1

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