Posted 14
March 2007
Subsets and
inclusion
Every
integer is a rational
number. This means that the sets 
and have a special relationship to each other: every element of is
an element of . This is the relationship
captured by the following definition.
For all sets A and B, if and only if every element of A is also an element of B.
This definition gives a rule of inference. The statement “ ” is read " A is included in B" or " A is a subset of B".
Examples
¨
.
¨
¨
The interval .
Terminology
Notation
The notation for inclusion has gotten in the last fifty years. The sad story is discussed in detail in the
section on symbols. Here
I give the three symbols used in abstractmath.org:
¨
means that every element of A is an element of B. In particular for any set
A, .
¨
(A is
a proper subset of B) means that every element
of A is an element of B, but there is at least one element of B that is not an element of A. For
example, because every integer is a real number but
there are real numbers that are not integers.
(See proper for some ambiguity in the use of this
word.)
¨
is the negation of . It means that there is at least one element
of A that is not an element of B. For
example, .
The word "contain"
is ambiguous as mathematicians usually use it.
¨
If you may say " A contains x".
¨
If you may say “A contains B”.
This can be confusing if the set A contains both sets and other things as elements. writers say “A contains x as an element”
and “A contains B as a
subset”.
Basic facts about inclusion
Fact For every set A, . In other words, every set is a subset of itself.
Fact If and then .
Fact For every set A, . Proof.
Warning
The statement that every set is a subset of itself can cause
cognitive
dissonance.
Assertions form subsets
If P(x) is an assertion
whose only variable is x then
the set of elements of a set S for
which P(x) is true is a subset of S. Using
setbuilder notation, this subset is .
Examples
¨
Let S =
{2, 3, 4, 5 ,6} and let P(n)
be the assertion “n is an even
integer”. Then P determines the subset {2, 4, 6} of S, and we could write the subset as .
¨
The circle of radius 1 with center at the origin
is the subset of the xy
plane.
¨
If x is
real, then is a subset of the set of real numbers . Of course it is the empty set.
¨
If x is
a complex variable, then the set is a subset of . Of course, it
is not
empty. In fact .