The answersDownload the PDF
Worksheet

A8.3 Boolean identities and laws

Boolean algebra and logic circuits · A level · OCR H446 1.4.3, AQA 7517 4.6.5.1, Eduqas A500QS 1.2 · about 20 min

BugBotLab
NameClassDate

What this lesson is about

The identities, commutation, association, distribution, double negation and absorption, proved by truth table.

Questions 6 marks in all

  1. [1 mark]Which law is used in the step A ∧ (B ∨ C) = (A ∧ B) ∨ (A ∧ C)?

    1. ADistribution
    2. BAssociation
    3. CCommutation
    4. DAbsorption

  2. [1 mark]What is A ∨ 1?

    1. A1
    2. BA
    3. C0
    4. D¬A

  3. [1 mark]What does A ∨ (A ∧ B) simplify to?

    1. AA
    2. BA ∧ B
    3. CA ∨ B
    4. DB

  4. [1 mark]What does A ∨ (¬A ∧ B) simplify to?

    1. AA ∨ B
    2. BA
    3. CB
    4. DA ∧ B

  5. [1 mark]Which of these are true for every value of A, B and C?

    Tick every answer that is true.

    1. AA ∧ ¬A = 0
    2. BA ∨ B = B ∨ A
    3. CA ∨ A = 2A
    4. DA ∧ (A ∨ B) = A
    5. EA ∨ (B ∧ C) = (A ∨ B) ∧ C

  6. [1 mark]What does this program print?

    count = 0
for a in [0, 1]:
    for b in [0, 1]:
        for c in [0, 1]:
            if (a | (b & c)) != ((a | b) & c):
                count = count + 1
print(count)

The task: prove it by brute force

Write equivalent(f, g). Its inputs f and g are functions of three bits, f(a, b, c), each returning 0 or 1. It tries all eight rows in binary order (a, then b, then c, each 0 then 1) and returns the first row where f and g give different results, as a tuple (a, b, c), or None if they agree on every row. Then, for each (name, f, g) in the list laws, in order, print one line: <name>: equivalent if they agree, or <name>: not equivalent at A=<a> B=<b> C=<c> using the row equivalent returned. That makes six lines.

# the two lines every program starts with: the commands, then the robot
from bugbot import *
connect()

laws = [
    ("distribution", lambda a, b, c: a & (b | c), lambda a, b, c: (a & b) | (a & c)),
    ("absorption", lambda a, b, c: a | (a & b), lambda a, b, c: a),
    ("association", lambda a, b, c: (a | b) | c, lambda a, b, c: a | (b | c)),
    ("OR over AND", lambda a, b, c: a | (b & c), lambda a, b, c: (a | b) & (a | c)),
    ("false friend", lambda a, b, c: a | (b & c), lambda a, b, c: (a | b) & c),
    ("half absorbed", lambda a, b, c: a & (a | b), lambda a, b, c: a | b),
]

def equivalent(f, g):
    return None

Plan your program here, then type it in and press Run.

QR code
Do it on the robot
www.bugbotlab.com/learn/a8-3-boolean-identities-and-laws/
The simulator checks it and tells you when it passes. Nothing to install, no account.

Challenges

  1. The "false friend" looks like association but mixes two operators. Explain in one sentence why the brackets matter there.
  2. Prove A ∧ (A ∨ B) = A by algebra, naming the law at each step.
  3. Add a check that f and g only ever return 0 or 1, and use it to catch a function written with ~.