Deductive reasoning refers to the process where one derives a conclusion (C) starting with a known (or assumed) set of premises (P). An example may illustrate this better:

P: Assume all men will die someday

P: Assume bin Laden is a man

C: bin Laden will die someday

This deductive step was based on the logical principle that if A implies B, and A is true, then B is true.

Deductive reasoning can fail. This is seen in the following famous paradox:

Let a = b Thus, a^{2}= ab a^{2}+ a^{2}= a^{2}+ ab 2a^{2}= a^{2}+ ab 2a^{2}- 2ab = a^{2}+ ab - 2ab 2a^{2}- 2ab = a^{2}- ab Rewrite this as: 2(a^{2}- ab) = 1(a^{2}- ab) Dividing both sides by (a^{2}- ab) gives: 2 = 1 QED.

See the problem with this line of reasoning? This is a very good example that one cannot apply deductive reasoning blindly for mathematical proofs.

Hint: Substitute the original assumption back into the numerator...

Another, much more subtle example is given below. See if you can find the error made in the reasoning:

Which reduces to 1 = 2 (Adding terms)

QED.

Hint: Do not make any assumptions about algebra other than those you have explicitly seen proofs for.