STEP-BY-STEP LESSON · §2.4
Application: Digital Logic Circuits
Connect a logical formula to circuit inputs and output.
Before you begin
Logical Form and Logical Equivalence
p and q stand for statements, each true or false. Compound formulas combine them with logical operations.
p∧q
p∧q is true only when both parts are true. p∨q is true when at least one part is true. ¬p reverses the value.
¬(p∧q)
Symbols
- 0/1
- false/true in a circuit
- ∧
- AND gate
- ∨
- OR gate
- ¬
- NOT gate
Definitions and notation for this topic
True / false
T / F; ⊤ / ⊥
T and F denote true and false. The symbols ⊤ and ⊥ are truth and falsity constants in the stated logical notation.
In digital logic, 1 and 0 encode these values. They do not make the logical operation OR the same as ordinary integer addition: 1∨1=1.
Separate glossary example
The statement 3 = 3 has truth value T.
Binary numerals and circuit values
(b_k…b_0)₂; 0/1
In a binary numeral each digit is 0 or 1 and the subscript 2 specifies place-value base two. In digital logic, 0 and 1 can instead encode false and true.
Distinguish arithmetic addition, 1+1=(10)₂, from OR. A binary numeral’s subscript is a base marker, not a sequence index.
Separate glossary example
(1011)₂=8+2+1=11. In a circuit, 1∧0=0 and 1∨1=1.
Step by step
Step 1 / 6
Input bits
Use 0 and 1 as two logical values in the model. Gates perform specified operations.
This is a mathematical model of a digital circuit.
Worked example
Inputs p=1,q=0,r=1 enter out=(p∧q)∨¬r. Trace both branches before the final OR.
- AND branch: a=1∧0=0, because AND needs both inputs to be 1.
- NOT branch: b=¬1=0, because NOT reverses the input bit.
- The final OR receives a=0,b=0, so out=0∨0=0. Neither branch supplies a 1.
Optional self-check
AND output for 1 and 1?
Show answer
1. Both inputs are true.