(related to Chapter: Logical Arguments Used in Mathematical Proofs)
If a conclusion is made that a "specific case must be true" since the general principle is known to be true, then we call that conclusion deductive reasoning. Deductive reasoning has the following logical form:
"$p$ is true, therefore, a special case of $p$ is true."
Deductive reasoning is valid, since the special case (conclusion) must be true, because it is an instance of a true general case (premise). Deductive reasoning is a fundamental way of showing the truth of a special case referring to known mathematical result. Examples of such reasoning are:
"All prime numbers are only divisible by $1$ and by themselves. $25$ is divisible by $1$, by $25$, and also by $5$. Therefore, $25$ is not a prime number."
"The sum of the angles in any triangle in a plane is 180 degrees, therefore the sum of the angles of a right-angled triangle in a plane is also 180 degrees."
"Any differentiable function is continuous $f(x)=3x$ is differentiable with the derivative $f'(x)=3.$ Therefore, $f$ is continuous."