# how to convert truth table into boolean expressions?

• Sep 11th 2009, 11:22 PM
Khonics89
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 ??
• Sep 12th 2009, 05:34 AM
Boolean Expression from a Truth Table
Hello Khonics89
Quote:

Originally Posted by Khonics89
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.

• 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.)

• Sep 12th 2009, 05:20 PM
Khonics89
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 ??
• Sep 12th 2009, 10:09 PM