Information technology

1. TRUTH TABLE
• Remember:
– Decimal to Binary
– Binary to Decimal
– Binary math
– Finish representation of data
• Numbers
• Symbols
• Images
• Audio
• Today:
– Boolean Logic
– Truth Tables
– Circuits
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3. What we will cover…
• The connection between Boolean logic and circuits
• Boolean (Propositional) Logic
• The design of simple logic circuits
• Representing simple logical sentences in hardware
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4. Logic and Logic Circuits
• What is a circuit?
– the complete path of an electric current, or a collection of electronic
elements
– We are interested in logic circuits, those whose output varies
depending on their input.
• Logic circuits emulate the Boolean logic operators
from propositional logic
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5. Binary for Logic?
• In Propositional Logic we examine the “truth” of
sentences.
– Sentences consist of:
• Variables:
– sub-sentences (A, B, C …) which are either true or false
– In computer science, we use
» true = 1
» false = 0
• Logical Operators:
– e.g., NOT, OR, AND, XOR, NAND, NOR
• Example sentence:
– X = (A AND B) OR C
• We examine the logic of a sentence through “truth table
analysis” 4-5

6. Simplest Truth Table
a Single Variable
• A single variable has only two possible values in
Boolean logic:
– true = 1
– false = 0 A
• A “truth table” represents all of the possible
values of a sentence given the possible values of 1
its inputs (variables).
– We determine the output by considering all possible truth
values for each variable
– we want to see the results of all possible combinations of
0
each variable
• How many rows should appear in a given truth
table? • Example: A = “It is raining” 4-6

7. Logical AND
• Logical AND:
– Takes two variables
– Evaluates to True only if both variables are true
– Written as: (A B)
• Example:
– A = It is raining
– B = I am in London
– (A B) = It is raining and I am in London
– What is the truth table for (A B) ?
• How many combinations of values exist for A AND B?
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8. Logical AND on two variables
It is raining and I
It is raining I am in London
am in London
A B A B
1 1 1
1 0 0
0 1 0
0 0 0
A AND B is true if both A is true and B is true
Otherwise, A AND B is false.
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9. Logical Or
• A OR B
• A B
A B A B
• Has two values: true if either A
or B is true, or if both A and B
1 1 1 are true
• false if they are both false.
1 0 1 • Are either of these things true?
– Note: both can be true… this is not
“exclusive or”
0 1 1
0 0 0
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10. Logical Not
• Used to invert a meaning
• NOT A as an alternative:
A A
• A = “It is raining”
1 0 • A = “It is not raining”
0 1
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11. XOR- Exclusive Or
A B A XOR B • True when either A or B are
true, but not both
1 1 0 • So “A and NOT B” or “B and
NOT A”
1 0 1 • Built from simpler logic:
(A B) ( A B)
0 1 1
0 0 0
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12. Nor and Nand
A B A B A B A Nand B A Nor B
1 1 1 1 0 0
1 0 0 1 1 0
0 1 0 1 1 0
0 0 0 0 1 1
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13. Truth Table Analysis
• How do you build a truth table?
– Step 1: Create columns for all the variables in the sentence
– Step 2: Determine the number of rows you need given the variables in your
sentence
– Step 3: Define all possible sequences (cases) for your truth table, starting
with all variables false and ending with all variables true
– Step 4: Deconstruct the logic in the sentence and fill in your table
– What is the truth table for:
1. X = A AND (NOT B)
2. X = (NOT A) AND (NOT B)
3. X = (A OR B) AND C
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14. Logic and Logic Circuits
• What is a circuit?
– the complete path of an electric current, or a collection of electronic
elements
– we will consider transistors to be the basic building blocks of logic
computer hardware.
– Logic circuits are built from a series of transistors
• What is a transistor?
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15. Transistors
– A transistor is an electronic device that
has three ends: a source, a sink, and a
source gate
– In this type of transistor, when the gate
is:
gate • ON, power flows from the source to
the sink.
• OFF, power does not flow to the
sink
sink
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16. Transistors (like faucets)
The operation of a transistor could be explained by making an analogy to faucets.
• A faucet has:
– An input
• the water company
• “source”
– An output
• a sink (where water is drained)
• “sink”
– Flow control
• If the tap knob (gate) is turned :
– ON water flows from the source to the sink
– OFF no water flows.
• The state of the tap determines the presence of water at the sink
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17. Transistors (like faucets)
gate gate
source source
OFF ON
sink sink
• Changing from water to electricity … in transistors:
– Electricity flows from the source to the sink with the gate = 1 (ON)
– Electricity does not flow from the source to the sink with the gate = 0 (OFF)
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18. Transistors
– The current technology used to build computer
hardware (chips) is called CMOS.
source • In CMOS we also use another kind of
transistor, distinguished by the little
bubble
• The bubble means that this transistor
works in the opposite way (it's ON
gate when the gate is OFF and OFF when
the gate is ON).
sink
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19. Building Complicated Circuits with
Transistors
Battery
• What on earth does this do?
If A=0 then . . .
bottom gate is off
top gate is on
power flows from the battery, can’t go out the sink,
and goes out through Z. Z = 1
If A = 1 then …
A Z bottom gate is on
top gate is off
power doesn’t get to Z from the battery, and any
power left in Z will flow out the sink. Z = 0
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20. Building Circuits with Transistors
Battery
A ?
1 0
A Z 0 1
What logical operator does this
circuit perform?
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