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Complement
Suppose we have a sample space
and some other set A which is a subset of
.
The set of all the elements of
which are NOT in A is called the complement of A,
and it is written Ac. Everything
in the sample space
is in either A or Ac, so
that
AAc
=
.
Together, the set and its complement cover all the possible outcomes.
But also
AAc
=
,
since the set A and its complement cannot have anything in common.
As an example, look again at the dice example; the sample space is
{1,2,3,4,5,6}. If A={1,2,3,6}, then Ac={4,5}. And the complement of {4,5} is {1,2,3,6} again;
the complement of the complement of some set is the original set.
This is written (Ac)c
.
For the tuberculosis example, if L is
the event that a person has pulmonary tuberculosis, and E is
the event that a person has extrapulmonary tuberculosis. Here,
the complement Lc is the event
that a person does not have pulmonary involvement; the complement
Ec is the event that a person
does not have extrapulmonary involvement.
Now
LcEc
is the event that a person is both a person who who does not have pulmonary
disease and also does not have extrapulmonary disease, or in other words,
that a person has neither disease--this is the complement of the union:
LcEc
=
( LE
)c.
This last equation is always true, for any sets L and E,
and is one of the two De Morgan Rules; the complement of the union
is the intersection of the complements.
Also
LcEc
is the event that a person either does not have extrapulmonary disease,
or does not have pulmonary disease (or possibly does not have both).
To be in this union, you have to be in at least one of the sets; so you
would have to be in either the group that does not have extrapulmonary
disease or the group that does not have pulmonary disease, or both. But
you could not have both.
LcEc
=
( LE
)c.
So the complement of the intersection is the union of the complements; this
is the other De Morgan Rule.
.
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