Tips for Proving Statements – Perfectly Logical

Here are some tips from me to you that I hope will help you prove statements or be a refresher on techniques that you have at your disposal. This is by no means an all-encompassing guide on proving. This is just my experience bottled up and sent via HTML ;-).

Dissecting a Statement

Math builds on itself. It is rigid and everything is clearly defined… or at least it should be. So, when you are presented with a statement that needs to be proved, read it. Note all the words that you don’t know. You should look up the definitions to these words. Then, if there are words in definitions of those words that you don’t know. Look them up. Repeat this process until you get to words that you know. Then, build yourself back up to the statement at hand. Think about what you need to prove and how what you know will help you prove the statement.

As a side note, this also works well in life situations.

Writing a Mathematical Proof

This is the structure that I use when I am attempting to prove a statement.

State what you know that is given to you by the statement. Then, if necessary, define variables and other objects from what you are given.
Continue to use what you know to arrive at the result of the statement, using theorems, definitions, lemmas, and so on.
Conclude your proof with “Therefore” followed with the proven statement.

Everyone proves differently but I try to make my proofs as efficient as possible. In my opinion, you should avoid flowery statements and mentioning things that don’t pertain to the proof or what is given to you. It is a waste of space and time for you and your reader.

Proof by Contradiction

There are some statements that are best proven by contradiction. To do this, you negate the statement then try to prove it. You know that the statement is false so there will be a contradiction as you try to prove the negated statement. Once you find the contradiction, you have proven the original statement.

Proof by Contrapositive

Much like proving by contradiction, but you negate the statement twice. By proving the doubly negated statement, you prove the original statement since, in mathematics, the contrapositive is equivalent to the original statement.

Mathematical Induction

There are two versions of mathematical induction: weak and strong. Technically, they are both the same but, in practice, they are different. Mathematical induction is useful in proving a statement when the statement iterates in someway that can be proven at each iteration. The following is summary of how to use weak induction which is a much more common technique of the two.

There are two steps to proving a statement using mathematical induction:

Prove the statement’s base case. The base case will be the lowest, or first, case that the statement is attempting to be proven at.
Assume the statement is true for k where n≤k. Then, prove that the statement is true for k+1. (You will use the assumption to prove this.)

Here is a short example of using mathematical induction to prove a statement involving Fibonacci numbers. Remember that Fibonacci numbers are described by the formula: F_n=F_n-1+F_n-2 where F₁=F₂=1.

Prove that F₁+F₃+F₅…+F_2n-1 = F_2n for n≥1.

Consider F₁+F₃+F₅…+F_2n-1=F_2n. Suppose n=1. Then, F₁=1 and F_2n=F₂₍₁₎=F₂=1. Hence, F₁+F₃+F₅…+F_2n-1=F_2n when n=1.

Suppose F₁+F₃+F₅…+F_2k-1=F_2k where n≤k. Consider F₁+F₃+F₅…+F_2k-1+F_2k+1. Then, F₁ + F₃ + F₅ + …+ F_2k-1 + F_2k+1= F_2k+F_2k+1=F_2k+2=F_2(k+1).

Therefore, F₁+F₃+F₅…+F_2n-1=F_2n for n≥1 by mathematical induction.

“If and only if” statements

An if and only if statement has two directions that need to be proven since if and only if statements are particularly strong statements. For example, saying, “a number is even if and only the number is divisible by 2” means two things. It means, if a number is even, it is divisible by 2 and it means that if a number is divisible by 2 the number is even. So, in order to prove such a statement, you must prove both statements to prove the original if and only if statement.

ε-δ Limit Proofs

Below is the precise definition of a limit.

Let f(x) be a function and x, a, L∈ℛ. Then, for $lim_{x \rightarrow a} f(x)=L$ , given any real number ε>0, there exists another real number δ>0 so that if 0<|x-a|<δ, then |f(x)-L|<ε.

In order to prove a limit using this definition, you will want to do some scratch work on the side. This will lead to finding a delta to choose. I think it would be best to see this with an example…

Prove that $lim_{x \rightarrow 1}4x+5=9$ .

First, we need to find a δ value that we will use later in the proof so that when we consider 0<|x-a|<δ, |(4x+5)-9|<ε. Your goal will be to simplify |(4x+5)-9| so that the term |x-1| is included in the expression: |(4x+5)-9|=|4x-4|=|4(x-1)|= |4||x-1|. Now, take what is not |x-1| and let δ equal ε divided by what is not |x-1| which is 4 in this case. So, we want to choose a δ= $\frac{ε}{4}$ . With our δ in hand, we are ready to write the formal proof.

Let ε∈ℛ, ε>0, and δ= $\frac{ε}{4}$ . Then, since ε>0 and δ= $\frac{ε}{4}$ , δ>0. Consider |x-1|< δ. So, |x-1|<δ= $\frac{ε}{4}$ , 4|x-1|<ε, |4x-4|<ε, and |4x+5-9|<ε.

Therefore, $lim_{x \rightarrow 1}{4x+5}=9$ .

In general, you will be doing some scratch work for ε-δ proofs to find a proper δ to choose.

Vacuous Proofs

A statement that can not be proven false is said to be true is called a vacuous statement.