3.4.2 - Logic circuits and Boolean expressions
Combine the four required gates, construct complete truth tables for up to three inputs, and translate between a circuit and a Boolean expression with the specified symbols.
Truth tables for combined gates
In a combined circuit, the output of one gate can become the input of another gate.
Suppose A and B enter an AND gate whose output is P. Input C enters a NOT gate whose output is Q. P and Q then enter an OR gate, producing Y. Calculate P and Q first; Y depends on their results rather than directly applying OR to A, B and C.
Three independent inputs have combinations. List them as 000, 001, 010, 011, 100, 101, 110 and 111. Extra columns make the calculation visible:
| A | B | C | P = A AND B | Q = NOT C | Y = P OR Q |
|---|---|---|---|---|---|
| 0 | 0 | 0 | 0 | 1 | 1 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 1 | 1 |
| 0 | 1 | 1 | 0 | 0 | 0 |
| 1 | 0 | 0 | 0 | 1 | 1 |
| 1 | 0 | 1 | 0 | 0 | 0 |
| 1 | 1 | 0 | 1 | 1 | 1 |
| 1 | 1 | 1 | 1 | 0 | 1 |
For inputs 101, P is 0 because B is 0; Q is 0 because C is 1; consequently Y is 0. Interpreting the full table, Y is on whenever C is 0, or when A and B are both 1. This is more informative than saying it is “on in five rows”.
Creating and modifying logic circuits
To draw a logic circuit, connect each gate's output to the next gate that needs that result.
The diagram below implements the circuit just calculated. A and B feed the AND gate, C feeds the NOT gate, and their outputs P and Q feed the final OR gate. Each wire carries a Boolean value, not a mathematical operation.
The two intermediate outputs are P = A AND B and Q = NOT C. The final gate computes Y = P OR Q.
Build separate gates for separate operations and check which side is an input and which is an output. A NOT gate acts on the particular wire attached to it. Putting NOT after the final OR gate would invert the complete result; it would not mean NOT C.
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