Boolean Logic Rapid-Review Notes

Boolean Logic – Core Ideas

• Computers use binary states $0$ / $1$ to represent FALSE / TRUE (Boolean logic).

Primary Logic Gates

• AND gate: output TRUE only if all inputs TRUE.
  • Truth table: \begin{array}{ccc}A & B & Q= A \land B\0 & 0 & 0\0 & 1 & 0\1 & 0 & 0\1 & 1 & 1\end{array}
• OR gate: output TRUE if any input TRUE.
  • Truth table: \begin{array}{ccc}A & B & Q= A \lor B\0 & 0 & 0\0 & 1 & 1\1 & 0 & 1\1 & 1 & 1\end{array}
• NOT gate: inverts a single input.
  • Truth table: \begin{array}{cc}A & Q= \lnot A\0 & 1\1 & 0\end{array}

Building Logic Circuits

• Combining gates forms a logic circuit.
• Standard symbols: AND (flat-front), OR (curved-front), NOT (triangle + small circle).

Writing Logic Expressions

1. Start with output letter and $=$ sign (e.g. $X =$ …).
2. Use brackets to show evaluation order (like maths).
3. Operators: $\land$ (AND), $\lor$ (OR), $\lnot$ (NOT).
   • Example: circuit with NOT on $A$ feeding OR with AND($B$,$C$): $X = (\lnot A) \lor (B \land C)$.

Drawing Circuits from Expressions

1. Evaluate innermost brackets first → draw those gates.
2. Connect resulting outputs into next operator’s gate.
3. End with overall output lead.

Truth Tables for Circuits

• List every possible input combination (for $n$ inputs: $2^n$ rows).
• Work left-to-right through the circuit, adding a column for each intermediate output.
• Final column = circuit output.

Quick Recall Tips

• AND = “all true”, OR = “at least one true”, NOT = “opposite”.
• Expression ↔ circuit translation relies on brackets and operator precedence.
• When stuck, build a truth table; identical output columns mean equivalent circuits/expressions.