Proof techniques in mathematical preliminaries is a sequence of mathematical statements that form an argument to show that a theorem is true.

The statements in the proof of a theorem include axioms (assumptions about the underlying mathematical structures), hypotheses of the theorem to be proved, and previously proved theorems. The main question is “How do we go about proving theorems?” This question is similar to the question of how to solve a given problem. Of course, the answer is that finding proofs, or solving problems, is not easy: otherwise life would be dull!

There is no specified way of coming up with a proof, but there are some generic strategies that could be of help. Here, we review some of these strategies. Here are some useful tips:

1. Read and completely understand the statement of the theorem to be proved. Most often this is the hardest part.
2. Sometimes, theorems contain theorems inside them. For example, “Property A if and only if propery B”, requires two statements:
   • If property A is true, then property B is true ( A ⇒ B).
   • If property B is true, then property A is true ( B ⇒ A).

   Another example is the theorem “Set A equals set B”. To prove this, we need to prove that A ⊆ B and B ⊆ A. That is, we need to show that each element of set A is i

3. Try to divide the theorems into multiple simple cases that either prove the facts or tautology of the case.
4. Try to write down the proofs once you have it. This is to ensure the correctness of your proof. Often, mistakes are found at the time of writing.
5. Finding proofs takes time, we do not come prewired to produce proofs. Be patient, think, express and write clearly and try to be precise as much as possible.

Proof Strategies

Direct proofs

Theorem 1: If n is an odd integer, then n² is odd as well.

Proof. An odd positive integer n can be written as n = 2k + 1, for some integer k ≥ 0. Then

n² = (2k+1)² = 4k² + 4k + 1 = 2(2k² + 2k) + 1.

Since (2k² + 2k) is even and “even plus one is odd”, we can conclude that n² is odd.

Theorem 2: Let G = ( V, E ) be a graph. Then sum of the degrees of all vertices is an even integer, i.e.,

∑_v∈V deg(v)

is even.

Proof. If you do not see the meaning of this statement, then first try it out for few graphs. The reason why the statement holds is very simple: Each edge contributes 2 to the summation (because an edge is incident on exactly two distinct vertices).

Constructive Proofs

This technique not only shows the existence of a certain object, it actually gives a method of creating it. Here is how a constructive proof looks like:

Theorem 4: There exists an object with property Ρ.

Proof. Here is the object: […]

And here is the proof that the object satisfies property Ρ: […]

Here is an example of a constructive proof. A graph is called 3-regular,  if each vertex has degree three.

Theorem 5: For every even integer n ≥ 4, there exists a 3-regular graph with n vertices.

Proof. Define

V = {0,1,2,3,….,n – 1},

and

E = { {i,i+1}: 0 ≤ i ≤ n – 2 } ∪ {{n-1,0}} ∪ {{i,i+n/2} : 0 ≤ i ≤ n/2 – 1 }.

Then the graph G = (V, E)  is 3-regular.

Convince yourself that this graph is indeed 3-regular. It may help to draw the graph for , say , n = 8 .

Nonconstructive proofs

In a nonconstructive proof, we show that a certain object exists, without actually creating it. Here is an example of such a proof:

Theorem 6: There exist irrational numbers x and y such that x^y is rational.

Proof. There are two possible cases.

Case 1: √2^√2 ∈ Q .

In this case, we take x = y = √2. In Theorem 9 Part 2 of this article, we will prove that √2 is irrational.

Case 2: √2^√2 ∉ Q.

In this case, we take x = √2^√2 and y = √2. Since

x^y = (√2^√2)^√2 = √2² = 2,

the claim in the theorem follows.

Observe that this proof indeed proves the theorem, but it does not give an example of a pair of irrational number and x and y such that x^y is rational.

Read Also: Mathematical Preliminaries of Theory of Computation