# 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, \$\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.

• 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.)

• 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