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When we refer to the cardinality of a finite set, we are talking about the number of distinct elements within that set. For example, if we were to count all the students in a classroom and find there are 25 students, the cardinality of the set representing the students is 25.

Under the umbrella of set theory, this concept is pivotal because it helps us understand sets quantitatively. But how do we know that a set cannot have two different cardinalities? Think of it this way: If you count all the apples in a basket, the number you end up with is the cardinality of your set of apples. Once you've counted them accurately, recounting will give you the same number—unless you add or take away some apples. This number is quintessentially unique to that specific configuration of the set. Theoretically, if someone counted differently, we would be discussing a fundamentally different set.

The demonstration of the uniqueness of cardinality can be rooted in the pigeonhole principle, which we'll explore in connection to functions next.

An injective function, also known as a one-to-one function, is a type of mapping between two sets where every element of the first set (the domain) is mapped to a unique element of the second set (the codomain). Put simply, no two different elements in the domain lead to the same element in the codomain. It's like having a party where everyone has to wear a unique hat—no duplicates allowed.

To illustrate, let's say we have two sets: Set A is the collection of students, and Set B is the collection of unique student IDs. An injective function would assign each student a different ID. If we were to find two students sharing an ID, that would violate the principle of injectiveness. It's important because this concept ensures that we can always trace back backwards from the codomain to the domain without confusion.

On the other hand, a surjective function, or an onto function, involves mapping every element from the domain to the codomain in such a way that every element in the codomain is 'hit' or 'covered' by the function. This means that for every element in the codomain, there is at least one element from the domain that maps to it.

Imagine we have a bucket of paint (set A) and a wall (set B). Using the paint to create paintings on the wall, a surjective function is akin to making sure every part of the wall gets painted—no spot should be left untouched. In terms of our previous analogy with students and IDs: if Set A is a list of student names and Set B is lockers, a surjective function from A to B would ensure every locker has at least one student assigned to it. A locker without a student would indicate we don't have a surjective function. Surjectiveness is crucial for knowing that our codomain is fully utilized without any spare, unassigned elements.