In this type of proof, a generalization is made by studying a single element of a set and its results are "induced" -- or persuaded -- over other elements of the same set. More precisely, this method of proof employs getting general results for a specific number of items from a larger set, and over those results, a generalization is made that all elements of the particular set will yield the similar results. For example, properties of all even numbers from a set can be generalized through induction by simply studying a single number and inducing its results over others.
This type of proof is at the foundation of set theory, which employs making an assumption and proving its results wrong in order to negate the assumption itself. In more general terminology, it presents an idea that if an assumption "A" results in "B" and "B" is wrong, then "A" is also wrong. The proof of contradiction is usually accompanied by many specifications and logical assumptions, through which a result is considered to be the reflection of its yielding problem.
Proof by contraposition encompasses a complex class of proofs that exist in set theory. In this type of proof, an assumption or a proposition is proved right or wrong by alternatively proving its contrapositive or inverse. More specifically, it can be imagined as a condition in which the relationship of set "A" and "B" can be proved by proving the relationship between "A-inverse" and "B-inverse." In other words, it's the method of verifying the proposition by proving its inverse or contrapositive in the similar scenario.
This is also one of the most common types of proofs established in set theory, and it verifies the assumptions made for a particular set through direct or straight logic. It's more like carrying out a theoretical derivation of a problem, which verifies its consecutive steps along with the results by simply referring to other established theorems, conjectures -- or estimations -- and proofs.