Boolean Expression Simplification: Truth Table, Circuit Diagram, and Optimization

To set up the truth table, we need to list out all the possible combinations of values for the variables A, B, and C, and then evaluate the expression for each combination.

| A | B | C | ~A | ~C | ~A ● B ● C | ~A ● B ● ~C | A ● ~B ● C | X | |---|---|---|----|----|-------------|-------------|-------------|---| | 0 | 0 | 0 | 1 | 1 | 0 | 0 | 0 | 0 | | 0 | 0 | 1 | 1 | 0 | 0 | 0 | 1 | 1 | | 0 | 1 | 0 | 1 | 1 | 0 | 1 | 0 | 0 | | 0 | 1 | 1 | 1 | 0 | 1 | 0 | 0 | 1 | | 1 | 0 | 0 | 0 | 1 | 0 | 0 | 0 | 0 | | 1 | 0 | 1 | 0 | 0 | 0 | 0 | 0 | 0 | | 1 | 1 | 0 | 0 | 1 | 0 | 1 | 0 | 0 | | 1 | 1 | 1 | 0 | 0 | 0 | 0 | 1 | 1 |

Now let's simplify the Boolean expression:

X = ~A ● B ● C + ~A ● B ● ~C + A ● ~B ● C

First, we can factor out B from the first two terms:

X = B ● (~A ● C + ~A ● ~C) + A ● ~B ● C

Next, we can factor out ~A from the first term:

X = B ● ~A ● (C + ~C) + A ● ~B ● C

Since C + ~C is always equal to 1, we can simplify further:

X = B ● ~A ● 1 + A ● ~B ● C

Now we can remove the unnecessary terms:

X = B ● ~A + A ● ~B ● C

Finally, let's draw the circuit:

                  ______
A ----|        NOT         |--------\
      |____________________|        |
                                     |   ______
B -----------------------------------|  | AND  |-----
                                      |________|     |
                                                      |
C --------------------------------------------------|

In the circuit diagram, the first NOT gate represents ~A, the second NOT gate represents ~B, and the AND gate represents the multiplication operation (●). The output of the AND gate is X.