Request about matlab code for verifying the de morgan s theorem

In propositional logic and boolean algebra, De Morgan's laws are a pair of transformation rules that are both valid rules of inference. They are named after Augustus De Morgan, a 19th-century British mathematician. The rules allow the expression of conjunctions and disjunctions purely in terms of each other via negation.

The rules can be expressed in English as:
the negation of a conjunction is the disjunction of the negations; and
the negation of a disjunction is the conjunction of the negations;
or
the complement of the union of two sets is the same as the intersection of their complements; and
the complement of the intersection of two sets is the same as the union of their complements.

In Boolean Algebra, there are some very important laws which are called the De Morgan's laws (the spelling can change from author to author).

These laws teach us how to interchange NOT with AND or OR logical operators.


In formal logic, De Morgan's laws are rules relating the logical operators 'and' and 'or' in terms of each other via negation:


not(p or q) =
(not p) and (not q)


not(p and q) =
(not p) or (not q)
Using logic gates (commonly used in Digital Electronics), they can be expressed in two forms:

the OR form:
De Morgan's laws

the AND form:
DeMorgan's law

In Matlab, these laws can be confirmed very easily.
Let's create a script file like this:


% Let x and y be column vectors
x = [0 0 1 1]'
y = [0 1 0 1]'

% We can confirm the OR form of the law with these two lines
x_or_y = x|y
DeMorg1 = not(not(x)& not(y))

% We can verify the AND form of the law with these two lines
x_and_y = x&y
DeMorg2 = not(not(x) | not(y))


When we run it, we get this output:

x =

0
0
1
1
y =

0
1
0
1

x_or_y =

0
1
1
1
DeMorg1 =

0
1
1
1
x_and_y =

0
0
0
1
DeMorg2 =

0
0
0
1

Which confirms the De Morgan's laws.