Proving Set Inclusion
While a Venn Diagram may be used
to illustrate set inclusion (or equality) its value is limited, usually to
three sets. Also, it would not be considered a proof by many. The usual
technique to show one set is a subset of another is a direct proof and
follows from the definition. If we wish to show X is a subset of Y,
we take a typical element 
and show that 
. For example,
suppose that for sets A and B, we wish to show that
We can argue as follows. Let x be an element of 
. Then x is
an element of A and an element of B. Certainly x is an element of
A and so is an element of A or B. Hence, x is an element of
. Since there was nothing special about x, it must be true for
all elements of . So by the definition of subset we conclude
.
We give the argument again, but this time using more symbols
and a brief explanation.
This is of course a simple example but does give the essential arguments.
As a slightly more difficult example, let's see how we would show
. This time we will tabulate
our proof. Note that the proof uses cases.
