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F9.2 Logic circuits and expressions

Logic and computer systems · GCSE · OCR J277 2.4.1, AQA 8525 3.4.2, Edexcel 1CP2 1.3.1 · about 15 min

BugBotLab

What this lesson is about

Combining gates, Boolean expressions, XOR, and the half adder.

Questions 6 marks in all

  1. [1 mark]In Q = (A AND B) OR (NOT C), A = 0, B = 1 and C = 1. What is Q?

    Answer: 0. A AND B is 0, NOT C is 0, and 0 OR 0 is 0.

  2. [1 mark]In Q = (A AND B) OR (NOT C), A = 0, B = 0 and C = 0. What is Q?

    Answer: 1. NOT C is 1, so the OR gives 1.

  3. [1 mark]When does XOR output 1?

    1. AWhen the inputs are different
    2. BWhen both inputs are 1
    3. CWhen at least one input is 1
    4. DWhen both inputs are 0
    Answer: A. Exclusive OR: one or the other, but not both.

  4. [1 mark]In a half adder, which gate gives the carry bit?

    1. AAND
    2. BXOR
    3. COR
    4. DNOT
    Answer: A. The carry is 1 only when both bits are 1.

  5. [1 mark]What does this program print?

    def XOR(a, b): return 1 if a != b else 0
for a, b in [(1, 1), (1, 0)]:
    print(a & b, XOR(a, b))
    Answer:
    1 0
0 1

    1 + 1 is carry 1, sum 0; 1 + 0 is carry 0, sum 1.

  6. [1 mark]"Beep if the robot is close and moving, or it has bumped." Which expression matches?

    1. A(close AND moving) OR bumped
    2. Bclose AND (moving OR bumped)
    3. C(close OR moving) AND bumped
    4. DNOT close AND moving AND bumped
    Answer: A. The and joins close and moving; the or adds bumped on its own.

The task: a half adder

Write AND(a, b) and XOR(a, b) as functions of 0s and 1s. XOR must be built from AND, OR and NOT functions you also write, as (A OR B) AND NOT (A AND B), not with !=. Print the half adder's truth table as four lines in the form A=1 B=1 carry=1 sum=0.

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

def AND(a, b): return 1 if a and b else 0
def OR(a, b): return 1 if a or b else 0
def NOT(a): return 1 - a

The hint students can ask for: XOR is true when the inputs differ, which is the same as saying at least one is true but not both: build it from OR, AND and NOT. The carry is the plain AND of the inputs.

A solution

from bugbot import *
connect()
def AND(a, b):
    return 1 if a == 1 and b == 1 else 0

def OR(a, b):
    return 1 if a == 1 or b == 1 else 0

def NOT(a):
    return 1 - a

def XOR(a, b):
    return AND(OR(a, b), NOT(AND(a, b)))

for a in [0, 1]:
    for b in [0, 1]:
        print(f"A={a} B={b} carry={AND(a, b)} sum={XOR(a, b)}")

Any program that meets the task's checks is marked correct in the simulator; this is one way, not the only way.