Define truth table and how to create one?

Answer 1

A truth table is used to simplify complex logic expressions, showing outputs for all possible inputs.

For the following examples, please forgive formatting. I've not yet learned how to make proper tables in plaintext.

Logical operations and their associated symbols:

* A OR B -- A + B

* A AND B -- A & B * A XOR B -- A ^ B * NOT A -- ~A * TRUE -- 1 * FALSE -- 0

A simple example with two variables. Evaluates the following boolean expression: A + B

A | B | A + B

----------------

0 | 0 | 0

0 | 1 | 1

1 | 0 | 1

1 | 1 | 1

Note how the first two columns correspond to the two inputs. Also note how the rows under those two columns show all possible combinations of inputs.

The next column shows our expression. In each row we use the inputs to find the output for the expression.

Of course, we could have figured all this out just by looking at the expression. So let's look at one that's a little more complex.

(A + B) & C

This one isn't as intuitive. So let's split it up. First we'll evaluate the A + B part the same way we did last time. Then we'll combine that information to evaluate the whole expression.

| A | B | C | A + B | (A + B) & C

|------------------------------------

| 0 | 0 | 0 | _ 0 _| 0

| 0 | 0 | 1 | _ 0 _| 0

| 0 | 1 | 0 | _ 1 _| 0

| 0 | 1 | 1 | _ 1 _| 1

| 1 | 0 | 0 | _ 1 _| 0

| 1 | 0 | 1 | _ 1 _| 1

| 1 | 1 | 0 | _ 1 _| 0

| 1 | 1 | 1 | _ 1 _| 1

So this expression will evaluate to true when given the inputs (0, 1, 1), (1, 0, 1), and (1, 1, 1).

By now you should be getting the hang of it. Use the earlier (simpler) parts of the truth table to evaluate the later (more complex) parts.

One more example for the sake of completeness.

~((A + B) ^ (B & C))

| A | B | C | A + B | B & C | (A + B) ^ (B & C) | ~((A + B) ^ (B & C))|

|---------------------------------------------------------------------------------|

| 0 | 0 | 0 | _ 0 _ | _ 0 _ | ______ 0 _____ | 1

| 0 | 0 | 1 | _ 0 _ | _ 0 _ | ______ 0 _____ | 1

| 0 | 1 | 0 | _ 1 _ | _ 0 _ | ______ 1 _____ | 0

| 0 | 1 | 1 | _ 1 _ | _ 1 _ | ______ 0 _____ | 1

| 1 | 0 | 0 | _ 1 _ | _ 0 _ | ______ 1 _____ | 0

| 1 | 0 | 1 | _ 1 _ | _ 0 _ | ______ 1 _____ | 0

| 1 | 1 | 0 | _ 1 _ | _ 0 _ | ______ 1 _____ | 0

| 1 | 1 | 1 | _ 1 _ | _ 1 _ | ______ 0 _____ | 1