[10.3c] Logic Expressions

Logic Expressions

Writing logic expressions from statements, circuits, and truth tables

A logic expression is a concise, symbolic way to describe how outputs depend on inputs using the operators NOT, AND, and OR (and, where appropriate, XOR, NAND, NOR). In this benchmark you will practise writing a correct logic expression when the description is presented as: a written problem statement, a drawn logic circuit, or a truth table. Expressions must accurately reflect the given description using the allowed inputs (maximum three) and a single output, typically labelled Q or X.

Throughout, think in three stages: identify the inputs and what they mean, decide which operator(s) link them, and place brackets to show the intended grouping. Brackets are essential for clarity. Remember that binary signals use 0 for false/low and 1 for true/high.

Language-to-operator quick guide

Wording in a statementTypical operatorExample mapping
"both", "at the same time", "all of"AND"both doors open" → A AND B
"either", "or", "at least one"OR"either sensor triggers" → A OR B
"not", "unless", "except when", "if ... is absent"NOT on that input"not locked" → ¬B
"exactly one", "one but not both"XOR"exactly one button pressed" → A ⊕ B or (A AND ¬B) OR (¬A AND B)

Worked pathways to expressions

Use the tabs to see how to produce a logic expression from each starting form. The examples are intentionally small (≤ 3 inputs) to mirror the exam constraints. Do not skip brackets: they show how the hardware or table structure translates into symbolic form.

Statement → Expression: "Alarm sounds when power is on AND (the door is closed OR the override is active)."
  1. Name inputs: A = power on, B = door closed, C = override active. Output Q = alarm.
  2. Spot keywords: "AND" joins main parts; "OR" joins the door/override choices. The phrase gives clear grouping: power AND (door OR override).
  3. Write the expression: Q = A AND (B OR C).
Tip: If you see "unless X", that usually means "AND ¬X". For example, "run the pump if A OR B, unless C" would be (A OR B) AND ¬C.
Contrast: Negation carefully placed

Wording: "Light is on when switch is on AND window is not open." → Q = A AND ¬B.

Wording: "Light is on when not (window open OR door open)." → Q = ¬(A OR B) (which is equivalent to ¬A AND ¬B, but you do not need to simplify here).

Mistake to avoid: Writing ¬A OR B when the statement really means ¬(A OR B). The position of NOT changes everything, so bracket carefully.

Circuit → Expression: read the wiring left to right

Given circuit description: Inputs A and B feed an AND gate. Input C passes through a NOT gate. The outputs of AND and NOT feed an OR gate to produce Q.

  1. Write the sub-results in order: X = A AND B, Y = ¬C.
  2. Combine for the output: Q = X OR Y.
  3. Substitute back: Q = (A AND B) OR (¬C).
Bracket discipline: Each gate corresponds to a bracketed sub-expression. Multi-stage circuits become nested brackets in the expression.
Variants and precise reading

If the diagram shows A, B, C entering a single 3-input AND gate, write Q = A AND B AND C. If built from cascaded 2-input AND gates, still write a single grouped AND.

If the final gate is NAND taking (A AND B) on its inputs, the expression is Q = ¬(A AND B). Keep the NOT outside the whole bracket to match the NAND symbol.

If a branch uses XOR with A and B, you may write A ⊕ B or expand as (A AND ¬B) OR (¬A AND B). Use whichever the exam accepts in your centre's notation.

Truth Table → Expression: sum-of-products or product-of-sums

Example table (two inputs): Q=1 when inputs are different.

ABQ
000
011
101
110
  • Sum-of-Products (SOP): make an AND term for each row where Q=1, using NOT for any 0, then OR the terms. Here: Q = (¬A AND B) OR (A AND ¬B).
  • Product-of-Sums (POS): make an OR term for each row where Q=0, again using NOT appropriately, then AND the terms. Here: Q = (A OR B) AND (¬A OR ¬B).
Which to use? SOP is usually quicker when there are few 1 rows; POS is handy when there are few 0 rows. Both are correct if built directly from the table.

General procedure for accuracy

  1. Define inputs clearly (e.g. sensors, switches). Stick to at most three inputs as per specification.
  2. Map words or symbols to operators and immediately decide where brackets are needed.
  3. Write helper sub-expressions for multi-stage logic, then merge them into a single final expression for Q.
  4. Cross-check by evaluating one or two rows you can predict (e.g. all 0s, all 1s) to see if the expression behaves sensibly.

Deep Dive: Negation placement and reading "unless"

Negation placement changes meaning. ¬(A OR B) means neither A nor B is true. (¬A) OR B means B is true or A is false, which is different. The word "unless" is often equivalent to "AND NOT": "start unless X" becomes start_condition AND ¬X. Brackets make these distinctions explicit and prevent ambiguity.

Terminology recap

  • Logic expression: a symbolic statement combining inputs with operators to produce Q.
  • Sum-of-products (SOP): OR of AND terms built from Q=1 rows.
  • Product-of-sums (POS): AND of OR terms built from Q=0 rows.
  • XOR: true when inputs differ; can be written explicitly or expanded as SOP.

 Key Takeaways

  • From statements, identify keywords and bracket groupings: write exactly what the wording implies.
  • From circuits, translate each gate into a bracketed sub-expression and combine in signal order.
  • From truth tables, build SOP from Q=1 rows or POS from Q=0 rows.
  • Negation placement is critical: ¬(A OR B) is not the same as (¬A) OR B.
  • Use at most three inputs and ensure one final output symbol, typically Q.