3.1 Logic Values, Gates, and Truth Tables

Chapter 2 showed how a bit pattern can represent a number or character. We now ask a different question: how does hardware transform one bit pattern into another?

Digital circuits answer with logic. A logic value is the abstract value 0 or 1. In a physical circuit, designers assign voltage ranges to these values; the exact voltages depend on the technology. At this level, we work with the reliable abstraction:

The words “low” and “high” describe signal ranges, not necessarily exactly 0 volts and 5 volts.

A logic gate applies one small rule

A logic gate is a circuit whose output is determined by a rule applied to its inputs. A NOT gate has one input. The gates below have two inputs, named A and B.

Here, Boolean $+$ means OR, not ordinary arithmetic addition. In particular, $1+1=1$ under the OR rule. The symbol $\oplus$ means exclusive OR.

A truth table checks every possible input

A truth table lists every input combination and its output. Two binary inputs have $2^2=4$ combinations:

A single example is never enough to identify a gate. AND and XOR both output 0 for input 00, but their other rows differ. A complete truth table is a small, exhaustive test.

Lab 1 — identify gates by behavior

Switch A and B, compare the highlighted truth-table row, and use the complete four-bit output signature to identify each rule.

Check your understanding

Gates can be connected into a larger rule

The output of one gate can become the input of another. Consider:

The circuit first asks whether A and B differ. It then allows that result through only when C is 1.

Three inputs have $2^3=8$ possible combinations. To verify the complete circuit:

1. Choose one row, such as A=1, B=0, C=1.

2. Compute the intermediate signal: $G=1\oplus0=1$.

3. Compute the final output: $Y=1\cdot1=1$.

4. Repeat for all eight rows.

This is a combinational circuit: once its input signals settle, its output depends only on the current inputs. It does not remember earlier inputs.

Lab 2 — build a circuit that passes all rows

Choose both gate stages and inspect every mismatch. The goal is not to make one lamp turn on by accident; it is to match the target function for all eight input combinations.

Check your understanding

Useful reasoning habits

• Name intermediate signals. Writing $G=A\oplus B$ separates two manageable steps.

• Test boundary rows. All-zero and all-one inputs often expose an incorrect gate quickly.

• Do not confuse XOR with OR. They differ at input 11: OR outputs 1, XOR outputs 0.

• Do not infer a circuit from one observation. Different circuits can agree on several rows and still implement different functions.

The next section packages these small rules into reusable combinational blocks: decoders select one destination, multiplexers select one source, and an arithmetic logic unit selects one operation.