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Math Help - how to convert truth table into boolean expressions?

  1. #1
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    how to convert truth table into boolean expressions?

    I was wondering if someone can show me a clean cut method to finding boolean expressions for any truth table..

    I'll show you an example...

    x y z | f(x,y,z)
    1 1 1 0
    1 1 0 1
    1 0 1 1
    1 0 0 1

    how would we go on about finding an expression for this ??
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  2. #2
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    Boolean Expression from a Truth Table

    Hello Khonics89
    Quote Originally Posted by Khonics89 View Post
    I was wondering if someone can show me a clean cut method to finding boolean expressions for any truth table..

    I'll show you an example...

    x y z | f(x,y,z)
    1 1 1 0
    1 1 0 1
    1 0 1 1
    1 0 0 1

    how would we go on about finding an expression for this ??
    I hope you realise that this is not a complete truth table for three input variables, x, y and z. You have only the four lines where x = 1; you need another four lines with x = 0 to complete the table.

    Nevertheless, to answer your question:

    • Look for the lines of the table where the output value is 1 (True)


    • For each of these lines, construct a logical expression, using AND and NOT as appropriate, to construct the expression that corresponds to the True-False values of the input variables. (See below for examples.)


    • Combine each of these expressions using OR's to give a single Boolean expression to represent the output of the complete table.


    • If desired (and if you can!), simplify the resulting Boolean expression.

    In the (part-)example that you quote, f = 1 on lines 2, 3 and 4.

    On line 2, x and y are True; z is False. So this line is x AND y AND (NOT z); or x \land y \land \neg z

    On line 3, x is True; y is False; z is True. So this line is x\land \neg y \land z

    Similarly line 4 is x \land \neg y \land \neg z

    Combining these three expressions using OR's then, we get:

    f =(x \land y \land \neg z) \lor (x\land \neg y \land z) \lor(x \land \neg y \land \neg z)

    (This could then be simplified to f = x \land (\neg y \lor \neg z), if your Boolean algebra is up to it.)

    Grandad
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  3. #3
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    thanks grandad- so our main aim is to write individual expressions for each row and then sum them up using "or"

    and then simplify if possible ??
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  4. #4
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    Quote Originally Posted by Khonics89 View Post
    thanks grandad- so our main aim is to write individual expressions for each row and then sum them up using "or"

    and then simplify if possible ??
    That's a brief summary of what I said, yes.

    Grandad
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