Download 2.4 - Nature & Capabilities of Software

Boolean Logic & Truth Tables
In today’s lesson we will look at:
• a reminder about truth values and
NOT, AND, OR and EOR
• truth tables
• operator precedence
• equivalent expressions
Truth Values
When we first looked at Boolean Logic, we
introduced the idea of truth values – things that
we could all agree on as being true or false.
In computing, we are usually concerned with
truths about circuits, e.g.
• the switch is on
• the light is on
or mathematical truths (especially is application
programming).
Mathematical Truths
• We have looked at Boolean variables, and also used
if to test the truth of mathematical comparisons.
Mathematical truths are usually easier to determine:
– 1 is equal to 1
– 2 is more than 5
– 3 is NOT equal to 5
TRUE
FALSE
TRUE
• They might involve a variable, e.g. if x = 10:
– x is equal to 1
FALSE
– x is more than 5
TRUE
– x is NOT equal to 5
TRUE
Truth Values
• Truth values are pairs of opposites, and are usually
written in one of three ways:
– On or Off
– True or False
– 1 or 0
• For the rest of this topic we are going to use 0 and 1 as
they are easier to write.
• Most programming languages also allow you to use
numbers as Boolean values, with 0 as False and non-zero
as True (and Python even allows you to use strings, where
“” is False and a non-blank string is True), e.g. if x: is the
same as if x > 0: (or if x: is the same as if x != “”:)
Truth Table - NOT
• We can use truth tables to describe the output of
a particular Boolean logic operation.
• Truth tables describe the output for all possible
inputs
• e.g. the NOT operator toggles the truth value to its
opposite value:
Inputs and outputs are
p
NOT p
0
1
1
0
labelled
NOT operates on a
single value, so there
are only two possible
inputs; 0 or 1
Truth Table - OR
The OR operator gives a true result if any
of the input values is true, e.g.
p
0
0
1
1
q
0
1
0
1
p OR q
0
1
1
1
OR
operates on
two values,
so there are
four
possible
inputs
Truth Table - AND
The AND operator gives a true result if
both of the input values are true, e.g.
p
0
0
1
1
q
0
1
0
1
p AND q
0
0
0
1
Truth Table - EOR
The EOR operator gives a true result if the
two input values are different, e.g.
p
0
0
1
1
q
0
1
0
1
p EOR q
0
1
1
0
Is Anything Always True?
The following always give the same results:
• p OR NOT p is always 1
p
0
1
NOT p
1
0
P OR NOT p
1
1
• p AND NOT p is always 0
p
0
1
NOT p
1
0
P AND NOT p
0
0
Combining Operators
• AND and OR can work with more than two
inputs, just take a pair of inputs at a time:
p
q
R
p OR q OR r
0
0
0
0
0
0
1
1
0
1
0
1
0
1
1
1
1
0
0
1
1
0
1
1
1
1
0
1
1
1
1
1
• Just like BIDMAS for arithmetic, there is a
correct order to combinations of AND and OR.
Operator Precedence
• There are rules about which operations should
be done first; this is known as operator
precedence:
– operations inside brackets are done first
– AND is done before OR
• In fact, sometimes a AND b is written as a.b
and a OR b can be written as a+b – this can
help you to remember the order, as
multiplication comes before addition in
BIDMAS for ordinary arithmetic.
Duality
• Look at the truth tables for AND and OR –
what do you see?
p
q
p AND q
p
q
p OR q
0
0
0
1
1
1
0
1
0
1
0
1
1
0
0
0
1
1
1
1
1
0
0
0
• If you swap 0 for 1 you get the other table!
• This is a property known as duality, and it can
be useful for simplifying calculations (or logic
circuits)
DeMorgan’s Duals
• This property leads to two rules, called
DeMorgan’s Duals:
– NOT(a OR b) = NOT a AND NOT b
– NOT(a AND b) = NOT a OR NOT b
• It’s unlikely that you will need to know these
for GCSE Computing, but they can be useful
for simplifying calculations or logic circuits.
• We will look at logic circuits and the symbols
used in them in the next lesson.