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 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 ??
Hello Khonics89I hope you realise that this is not a complete truth table for three input variables, $\displaystyle x, y$ and $\displaystyle z$. You have only the four lines where $\displaystyle x = 1$; you need another four lines with $\displaystyle x = 0$ 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, $\displaystyle f = 1$ on lines 2, 3 and 4.
On line 2, $\displaystyle x$ and $\displaystyle y$ are True; $\displaystyle z$ is False. So this line is $\displaystyle x$ AND $\displaystyle y$ AND (NOT $\displaystyle z$); or $\displaystyle x \land y \land \neg z$
On line 3, $\displaystyle x$ is True; $\displaystyle y$ is False; $\displaystyle z$ is True. So this line is $\displaystyle x\land \neg y \land z$
Similarly line 4 is $\displaystyle x \land \neg y \land \neg z$
Combining these three expressions using OR's then, we get:
$\displaystyle f =(x \land y \land \neg z) \lor (x\land \neg y \land z) \lor(x \land \neg y \land \neg z)$
(This could then be simplified to $\displaystyle f = x \land (\neg y \lor \neg z)$, if your Boolean algebra is up to it.)
Grandad