Derive function from circuit and circuit from function from Boolean algebra

Boolean algebra allows a fascinating interplay between logical expressions and physical circuits. Here's how we can go back and forth between them:

Deriving Logic from Circuits:

Identify the components: Recognize basic logic gates like AND,  OR,  NOT,  XOR,  etc used in the circuit.
Trace signal flow: Follow the path of signals through the circuit, observing how they interact with each gate.
Write Boolean expressions: For each gate, write an expression representing its output based on its inputs and operation (e.g., AND = A ∧ B).
Combine expressions: Connect the expressions of individual gates based on how they connect in the circuit, using parentheses for clarity.
Simplify (optional): Apply Boolean algebra rules (e.g., De Morgan's Law) to simplify the final expression if possible.

Example: Consider a circuit with two inputs (A, B) and an AND gate followed by an OR gate connected to both outputs of the AND gate. The logic derived from this circuit would be: F = (A ∧ B) ∨ (A ∧ B) which simplifies to F = A ∧ B.

Building Circuits from Logic:

Analyze the expression: Understand the variables and the operations involved in the Boolean expression.
Break down into sub-expressions: Identify simpler operations within the expression that can be implemented by individual gates.
Choose appropriate gates: Assign each sub-expression to a suitable logic gate (e.g., AND for A ∧ B).
Connect the gates: Arrange the gates based on the order of operations in the expression, connecting their inputs and outputs appropriately.

Example: To build a circuit for F = A ∨ (B ∧ C), you would connect A and B directly to an OR gate. The output of the OR gate would be connected along with C to another AND gate. Finally, the output of the AND gate would be the final output of the circuit.