UMBC CMSC203, Discrete Structures, Spring 2009
An existence proof establishes the existence of a mathematical object with
some properties we desire. Existence proofs may be constructive or
non-constructive. A constructive proof explicitly specifies the object.
Here, the specification can be mathematical --- infinite objects do not
have to be written down. Non-constructive existence proofs only show that
the desired object exists. Some examples will clarify this concept.
Example 1: In the section on proof by contradiction, we used
a graph that was not 3-colorable. This graph has triangles --- 3 vertices
that are connected to each other. For example, vertices a, e and c form a
It was very useful to have triangles in our argument that the graph is not
3-colorable, because the 3 vertices in a triangle have to be colored with 3
different colors. Are triangles necessary to prove that a graph is not
Claim: There exists a triangle-free graph that is
4-colorable, but not 3-colorable.
Consider the following graph and its 4-coloring:
By checking every triple of vertices, the reader can verify that
the graph is triangle-free. We leave the proof that the graph is not
3-colorable as an exercise.
Example 2: The proof in the previous example is definitely
constructive. The graph with the desired property is given explicitly. In
this example we prove the existence of a number with certain properties
without saying which number has the property.
Claim: There exist irrational numbers x and y such that
xy is rational.
Proof: Let z =
If z is rational then z is our desired number with
x = √2
y = √2.
Now, suppose that z is irrational. Then, let
x = z and
y = √2.
In this case, xy is again rational.
In either case, whether z is rational or irrational, we've
shown the existence of irrational numbers x and y
such that xy is rational.
The proof in Example 2 is definitely non-constructive.
The proof establishes the claim but does not tell us if it is
that is rational. Generally speaking, non-constructive proofs
are less satisfying than constructive proofs. In computer science,
especially, constructive existence proofs are preferred because we
often want to use the desired object in a program or algorithm.
A non-constructive proofs would not help there.
12 Feb 2009 12:55:13 EST
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