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While this may be a simplistic example, you can see that there’s no logical procession in a circular
argument.

Assuming the Truth of the Converse

Simply put: The fact that A implies B doesn’t not necessarily mean that B implies A. For example, “All
dogs are mammals; therefore, all mammals are dogs.”

Assuming the Truth of the Inverse

Watch out for this one. You cannot automatically assume the inverse of a given statement is true.
Consider the following true statement:

If you grew up in Minnesota , you’ve seen snow.

Now, notice that the inverse of this statement is not necessarily true:

If you didn’t grow up in Minnesota , you’ve never seen snow.

Faulty Generalizations

This mistake (also known as inductive fallacy) can take many forms, the most common being assuming a
general rule based on a specific instance: (“Bridge is a hard game; therefore, all card games are difficult.”)
Be aware of more subtle forms of faulty generalizations.

Faulty Analogies

It’s a mistake to assume that because two things are alike in one respect that they are necessarily alike in
other ways too. Consider the faulty analogy below:

People who absolutely have to have a cup of coffee in the morning to get going are as bad as alcoholics
who can’t cope without drinking.

False (or tenuous) analogies are often used in persuasive arguments.

Now that we've gone over some common mathematical mistakes, let's look at some correct and effective
ways to use mathematical reasoning.

Prove It

Let's look at basic logic, its operations, some fundamental laws, and the rules of logic that help us prove
statements and deduce the truth. First off, there are two different styles of proofs: direct and indirect.

Whether it's a direct or indirect proof, the engine that drives the proof is the if-then structure of a logical
statement. In formal logic, you'll see the format using the letters p and q, representing statements, as in:

If p, then q

An arrow is used to indicate that q is derived from p, like this: