Week 6 · Chapter 7 Computers: computer configuration, the five units, software, logic circuits and the half adder · 3 lessons
3
Logic gates: how a circuit adds two bits
ICT Secondary 1 Term 1, Chapter 7: pp. 85–94
GoalRead the truth tables of AND, OR, NOT and XOR, and explain how two gates add two bits.
New words
logic circuit · a circuit built from gates that work on 0 and 1truth table · a table of all inputs and the output of eachXOR · true when exactly one of the two inputs is 1half adder · a circuit that adds two bits and gives a sum and a carry
Explanation
A logic gate takes bits as inputs and gives one bit. AND gives 1 only if both inputs are 1: a door opens only if the card is valid AND the code is right. OR gives 1 if at least one input is 1: an alarm rings if there is smoke OR heat. NOT flips its single input: 0 becomes 1 and 1 becomes 0. XOR gives 1 if exactly one input is 1: a staircase light that two switches control is on when the switches are in different positions. A truth table lists every input and the output. A half adder adds two bits A and B. Check the four cases: 0 + 0 = 00, 0 + 1 = 01, 1 + 0 = 01 and 1 + 1 = 10 in binary. The right bit, the sum, is 1 exactly when one input is 1, so sum = A XOR B. The left bit, the carry, is 1 only when both are 1, so carry = A AND B. Two half adders and an OR gate make a full adder, which also takes a carry from the column on the right; chains of full adders add whole numbers inside the processor. Note that XOR can be built from AND, OR and NOT, but it is a gate of its own.
A
B
A∧B
A∨B
A⊕B
0
0
0
0
0
0
1
0
1
1
1
0
0
1
1
1
1
1
1
0
Worked examples
A
B
A⊕B
A∧B
0
0
0
0
0
1
1
0
1
0
1
0
1
1
0
1
The picture above shows AND (∧), OR (∨) and XOR (⊕) for every pair of inputs. Now the half adder: the sum column is the XOR and the carry column is the AND. For 1 + 1 the sum is 0 and the carry is 1, which together read 10, the binary number two.
smoke
heat
silenced
alarm
siren
1
0
1
1
0
1
0
0
1
1
0
0
0
0
0
NOT flips a bit. A small circuit: “alarm = smoke OR heat” and “siren = alarm AND NOT silenced”. If smoke = 1, heat = 0 and silenced = 1, then alarm = 1 and NOT silenced = 0, so siren = 1 AND 0 = 0.
Class activity
On paper, make the truth table for the invented rule “the garden light is on if it is dark AND someone is near”. Then find two input rows where the light is off although one input is 1.
Practice
1
What does OR give when A = 0 and B = 1?
1.
2
What does XOR give when A = 0 and B = 0?
0, because no input is 1.
3
Add the bits 0 + 1 with a half adder. What are the sum and the carry?
