Session Objectives#5
COULD construct a truth tables from a given logic diagram
SHOULD understand and produce simple logic diagrams using the operations NOT,
AND and OR
MUST explain why data is represented in computer systems in binary
Create a program using the LMC to calculate the perimeter of any
given quadrilateral. Try to design it so that it gives a running total.
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Binary Logic
Starter:
A lily pad doubles in size everyday. It takes 30 whole days to
fill up the whole pond, how many days did it take to fill half
the pond? Prove your answer...
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Binary Logic
We know that from von Neumann and the principle that all modern
computers, data and instructions are based on the binary system
(base 2). This is due to the ease in which 2 states can
recognised – 0 and 1, on and off, true or false – by using
simple transistors and capacitors.
transistor
capacitor
Memory uses very small transistors and capacitors which can be
linked together to make simple logical calculations: e.g are
both inputs 1? or is only one input 1? These simple circuits are
called Logic Gates.
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Logic Gates
There main gates are as follows:
1. NOT gate – it outputs the opposite of the input i.e input =
1, then output = 0, and vice versa.
Truth Tables are used to express the relationship between input
and output. (Algebraic values are used, ABC etc for input and
PQR for output)
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Input
Output
A
P
0
1
1
0
Logic Gates
2. AND gate – this tells us if both inputs are 1 by outputting
1, otherwise the output will be 0
e.g
A
B
P
0
0
0
0
1
0
1
0
0
1
1
1
3. OR gate – shows that either 1 OR 2 inputs are on by
outputting 1, otherwise output is 0.
e.g
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A
B
P
0
0
0
0
1
1
1
0
1
1
1
1
Logic Gate Diagrams
Each gate is represented by a different symbol:
INPUT
NOT gate
AND gate
OR gate
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OUTPUT
Logic Circuits
Logic gates can be joined together to make more complex logic
circuits.
A common combination is the NAND circuit (Not AND) which
frustratingly is a AND followed by a NOT gate. Similarly a NOR
is an OR followed by a NOT.
NAND – basically toggles the AND so that if both inputs are 1
then 0 will be output, otherwise 1 is output.
Output R
Output P
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A
B
R= A AND B
P=NOT R
0
0
0
1
0
1
0
1
1
0
0
1
1
1
1
0
Logic Circuits
This example has 3 inputs, 2 in the AND (A&B), outputting to an
OR at P, and 1 directly into the OR.
P
The resulting truth table is calculated:
A
B
C
P=A AND B
Q = P OR C
0
0
0
0
0
0
0
1
0
1
0
1
0
0
0
0
1
1
0
1
1
0
0
0
0
1
0
1
0
1
1
1
0
1
1
1
1
1
1
1
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Boolean Algebra
These logic circuits can be written down using mathematical
expersions called Boolean algebra (named after Mathematician
George Boole).
i.e Q = (A AND B) OR C
TASKS – Draw logic circuits and truth tables for the following
a) P=NOT(A AND B)
b) P=NOT(A OR B)
c) P=A AND NOT (B)
d) A AND NOT(B OR C)
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Boolean Programming
Boolean algebra is used in programming to perform many
instruction. For example IF statements and While loops
IF x >10 then...
ELSE....
__________________
WHILE x < 10 AND NOT (end of file) DO
Now try some simple programming using Ifs and Loops in Yousrc.
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